math.sin on complex, imaginary part

Percentage Accurate: 53.5% → 99.8%

Time: 11.8s

Alternatives: 21

Speedup: 2.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 0.04\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 + {im}^{5} \cdot -0.008333333333333333\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1.0) (not (<= t_0 0.04)))
     (* (* 0.5 (cos re)) t_0)
     (*
      (cos re)
      (-
       (+
        (* (pow im 3.0) -0.16666666666666666)
        (* (pow im 5.0) -0.008333333333333333))
       im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 0.04)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * (((pow(im, 3.0) * -0.16666666666666666) + (pow(im, 5.0) * -0.008333333333333333)) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-1.0d0)) .or. (.not. (t_0 <= 0.04d0))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * ((((im ** 3.0d0) * (-0.16666666666666666d0)) + ((im ** 5.0d0) * (-0.008333333333333333d0))) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 0.04)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * (((Math.pow(im, 3.0) * -0.16666666666666666) + (Math.pow(im, 5.0) * -0.008333333333333333)) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -1.0) or not (t_0 <= 0.04):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * (((math.pow(im, 3.0) * -0.16666666666666666) + (math.pow(im, 5.0) * -0.008333333333333333)) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -1.0) || !(t_0 <= 0.04))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) + Float64((im ^ 5.0) * -0.008333333333333333)) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -1.0) || ~((t_0 <= 0.04)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((((im ^ 3.0) * -0.16666666666666666) + ((im ^ 5.0) * -0.008333333333333333)) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1.0], N[Not[LessEqual[t$95$0, 0.04]], $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] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -1 or 0.0400000000000000008 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   if -1 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.0400000000000000008

   1. Initial program 9.4%

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

      1. neg-sub0 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in im around 0 99.8%

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

   5. Taylor expanded in im around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. neg-mul-1 99.8%

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

      3. unsub-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. associate-*r* 99.8%

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

      6. distribute-rgt-out 99.8%

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

      7. *-commutative 99.8%

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

      8. distribute-lft-out-- 99.8%

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

      9. *-commutative 99.8%

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

      10. *-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \lor \neg \left(e^{-im} - e^{im} \leq 0.04\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 + {im}^{5} \cdot -0.008333333333333333\right) - im\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1.0) (not (<= t_0 2e-11)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 2e-11)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-1.0d0)) .or. (.not. (t_0 <= 2d-11))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 2e-11)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -1.0) or not (t_0 <= 2e-11):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -1.0) || !(t_0 <= 2e-11))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -1.0) || ~((t_0 <= 2e-11)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1.0], N[Not[LessEqual[t$95$0, 2e-11]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -1 or 1.99999999999999988e-11 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   if -1 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1.99999999999999988e-11

   1. Initial program 8.0%

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

      1. neg-sub0 8.0%

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

   3. Simplified 8.0%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. distribute-rgt-out-- 99.8%

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

      6. *-commutative 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \cos re\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (cos re))))
   (if (<= t_0 -1.0)
     (* t_1 t_0)
     (*
      t_1
      (+
       (* im -2.0)
       (+
        (* -0.3333333333333333 (pow im 3.0))
        (+
         (* -0.016666666666666666 (pow im 5.0))
         (* -0.0003968253968253968 (pow im 7.0)))))))))

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 <= -1.0) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * pow(im, 3.0)) + ((-0.016666666666666666 * pow(im, 5.0)) + (-0.0003968253968253968 * pow(im, 7.0)))));
	}
	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 <= (-1.0d0)) then
        tmp = t_1 * t_0
    else
        tmp = t_1 * ((im * (-2.0d0)) + (((-0.3333333333333333d0) * (im ** 3.0d0)) + (((-0.016666666666666666d0) * (im ** 5.0d0)) + ((-0.0003968253968253968d0) * (im ** 7.0d0)))))
    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 <= -1.0) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * Math.pow(im, 3.0)) + ((-0.016666666666666666 * Math.pow(im, 5.0)) + (-0.0003968253968253968 * Math.pow(im, 7.0)))));
	}
	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 <= -1.0:
		tmp = t_1 * t_0
	else:
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * math.pow(im, 3.0)) + ((-0.016666666666666666 * math.pow(im, 5.0)) + (-0.0003968253968253968 * math.pow(im, 7.0)))))
	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 <= -1.0)
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(t_1 * Float64(Float64(im * -2.0) + Float64(Float64(-0.3333333333333333 * (im ^ 3.0)) + Float64(Float64(-0.016666666666666666 * (im ^ 5.0)) + Float64(-0.0003968253968253968 * (im ^ 7.0))))));
	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 <= -1.0)
		tmp = t_1 * t_0;
	else
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * (im ^ 3.0)) + ((-0.016666666666666666 * (im ^ 5.0)) + (-0.0003968253968253968 * (im ^ 7.0)))));
	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[LessEqual[t$95$0, -1.0], N[(t$95$1 * t$95$0), $MachinePrecision], N[(t$95$1 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   if -1 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 41.3%

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

      1. neg-sub0 41.3%

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

   3. Simplified 41.3%

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

   4. Taylor expanded in im around 0 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)\\ \end{array} \]

Alternative 4: 96.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (<= t_0 -1.0)
     (* (* 0.5 (cos re)) t_0)
     (*
      (cos re)
      (+
       (+
        (* (pow im 5.0) -0.008333333333333333)
        (* (pow im 7.0) -0.0001984126984126984))
       (- (* (pow im 3.0) -0.16666666666666666) im))))))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if (t_0 <= (-1.0d0)) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * ((((im ** 5.0d0) * (-0.008333333333333333d0)) + ((im ** 7.0d0) * (-0.0001984126984126984d0))) + (((im ** 3.0d0) * (-0.16666666666666666d0)) - im))
    end if
    code = tmp
end function

Java

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

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if t_0 <= -1.0:
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * (((math.pow(im, 5.0) * -0.008333333333333333) + (math.pow(im, 7.0) * -0.0001984126984126984)) + ((math.pow(im, 3.0) * -0.16666666666666666) - im))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   if -1 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 41.3%

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

      1. neg-sub0 41.3%

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

   3. Simplified 41.3%

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

   4. Taylor expanded in im around 0 98.9%

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

   5. Taylor expanded in im around 0 98.9%

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

      1. associate-+r+ 98.9%

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

      2. +-commutative 98.9%

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

      3. associate-*r* 98.9%

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

      4. neg-mul-1 98.9%

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

      5. distribute-lft-neg-in 98.9%

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

      6. distribute-rgt-out 98.9%

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

      7. associate-*r* 98.9%

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

      8. associate-*r* 98.9%

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

      9. distribute-rgt-out 98.9%

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

      10. distribute-lft-out 98.9%

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

      11. unsub-neg 98.9%

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

      12. *-commutative 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 99.2%

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

Alternative 5: 95.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ t_1 := e^{-im} - e^{im}\\ t_2 := 0.5 \cdot t_1\\ \mathbf{if}\;im \leq -5.7:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.029:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+25}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(t_1 \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 7.0) (* (cos re) -0.0001984126984126984)))
        (t_1 (- (exp (- im)) (exp im)))
        (t_2 (* 0.5 t_1)))
   (if (<= im -5.7)
     t_0
     (if (<= im 0.029)
       (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
       (if (<= im 3.6e+14)
         t_2
         (if (<= im 1.05e+25)
           (* (* re re) (* t_1 -0.25))
           (if (<= im 1.1e+44) t_2 t_0)))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 7.0) * (cos(re) * -0.0001984126984126984);
	double t_1 = exp(-im) - exp(im);
	double t_2 = 0.5 * t_1;
	double tmp;
	if (im <= -5.7) {
		tmp = t_0;
	} else if (im <= 0.029) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 3.6e+14) {
		tmp = t_2;
	} else if (im <= 1.05e+25) {
		tmp = (re * re) * (t_1 * -0.25);
	} else if (im <= 1.1e+44) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (im ** 7.0d0) * (cos(re) * (-0.0001984126984126984d0))
    t_1 = exp(-im) - exp(im)
    t_2 = 0.5d0 * t_1
    if (im <= (-5.7d0)) then
        tmp = t_0
    else if (im <= 0.029d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 3.6d+14) then
        tmp = t_2
    else if (im <= 1.05d+25) then
        tmp = (re * re) * (t_1 * (-0.25d0))
    else if (im <= 1.1d+44) then
        tmp = t_2
    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 t_1 = Math.exp(-im) - Math.exp(im);
	double t_2 = 0.5 * t_1;
	double tmp;
	if (im <= -5.7) {
		tmp = t_0;
	} else if (im <= 0.029) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 3.6e+14) {
		tmp = t_2;
	} else if (im <= 1.05e+25) {
		tmp = (re * re) * (t_1 * -0.25);
	} else if (im <= 1.1e+44) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 7.0) * (math.cos(re) * -0.0001984126984126984)
	t_1 = math.exp(-im) - math.exp(im)
	t_2 = 0.5 * t_1
	tmp = 0
	if im <= -5.7:
		tmp = t_0
	elif im <= 0.029:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 3.6e+14:
		tmp = t_2
	elif im <= 1.05e+25:
		tmp = (re * re) * (t_1 * -0.25)
	elif im <= 1.1e+44:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 7.0) * Float64(cos(re) * -0.0001984126984126984))
	t_1 = Float64(exp(Float64(-im)) - exp(im))
	t_2 = Float64(0.5 * t_1)
	tmp = 0.0
	if (im <= -5.7)
		tmp = t_0;
	elseif (im <= 0.029)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 3.6e+14)
		tmp = t_2;
	elseif (im <= 1.05e+25)
		tmp = Float64(Float64(re * re) * Float64(t_1 * -0.25));
	elseif (im <= 1.1e+44)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 7.0) * (cos(re) * -0.0001984126984126984);
	t_1 = exp(-im) - exp(im);
	t_2 = 0.5 * t_1;
	tmp = 0.0;
	if (im <= -5.7)
		tmp = t_0;
	elseif (im <= 0.029)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 3.6e+14)
		tmp = t_2;
	elseif (im <= 1.05e+25)
		tmp = (re * re) * (t_1 * -0.25);
	elseif (im <= 1.1e+44)
		tmp = t_2;
	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]}, Block[{t$95$1 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * t$95$1), $MachinePrecision]}, If[LessEqual[im, -5.7], t$95$0, If[LessEqual[im, 0.029], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.6e+14], t$95$2, If[LessEqual[im, 1.05e+25], N[(N[(re * re), $MachinePrecision] * N[(t$95$1 * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -5.70000000000000018 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.3%

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

   5. Taylor expanded in im around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   7. Simplified 98.3%

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

   if -5.70000000000000018 < im < 0.0290000000000000015

   1. Initial program 9.4%

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

      1. neg-sub0 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in im around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

      4. associate-*r* 99.5%

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

      5. distribute-rgt-out-- 99.5%

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

      6. *-commutative 99.5%

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

   6. Simplified 99.5%

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

   if 0.0290000000000000015 < im < 3.6e14 or 1.05e25 < im < 1.09999999999999998e44

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 86.0%

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

   if 3.6e14 < im < 1.05e25

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 100.0%

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

      4. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.7:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq 0.029:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+25}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(e^{-im} - e^{im}\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 6: 95.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -5.7:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.042:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 7.0) (* (cos re) -0.0001984126984126984))))
   (if (<= im -5.7)
     t_0
     (if (<= im 0.042)
       (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
       (if (<= im 1.1e+44)
         (* (- (exp (- im)) (exp im)) (+ 0.5 (* -0.25 (* re re))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = pow(im, 7.0) * (cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -5.7) {
		tmp = t_0;
	} else if (im <= 0.042) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = (exp(-im) - exp(im)) * (0.5 + (-0.25 * (re * re)));
	} 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 <= (-5.7d0)) then
        tmp = t_0
    else if (im <= 0.042d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 1.1d+44) then
        tmp = (exp(-im) - exp(im)) * (0.5d0 + ((-0.25d0) * (re * re)))
    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 <= -5.7) {
		tmp = t_0;
	} else if (im <= 0.042) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = (Math.exp(-im) - Math.exp(im)) * (0.5 + (-0.25 * (re * re)));
	} 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 <= -5.7:
		tmp = t_0
	elif im <= 0.042:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 1.1e+44:
		tmp = (math.exp(-im) - math.exp(im)) * (0.5 + (-0.25 * (re * re)))
	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 <= -5.7)
		tmp = t_0;
	elseif (im <= 0.042)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 1.1e+44)
		tmp = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	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 <= -5.7)
		tmp = t_0;
	elseif (im <= 0.042)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 1.1e+44)
		tmp = (exp(-im) - exp(im)) * (0.5 + (-0.25 * (re * re)));
	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, -5.7], t$95$0, If[LessEqual[im, 0.042], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.70000000000000018 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.3%

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

   5. Taylor expanded in im around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   7. Simplified 98.3%

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

   if -5.70000000000000018 < im < 0.0420000000000000026

   1. Initial program 9.4%

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

      1. neg-sub0 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in im around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

      4. associate-*r* 99.5%

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

      5. distribute-rgt-out-- 99.5%

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

      6. *-commutative 99.5%

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

   6. Simplified 99.5%

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

   if 0.0420000000000000026 < im < 1.09999999999999998e44

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 21.5%

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

      1. +-commutative 21.5%

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

      2. associate-*r* 21.5%

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

      3. distribute-rgt-out 79.4%

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

      4. unpow2 79.4%

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

   6. Simplified 79.4%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.7:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq 0.042:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 7: 87.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;im \leq -0.0102:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00105:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+165}:\\ \;\;\;\;\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im)))))
   (if (<= im -0.0102)
     t_0
     (if (<= im 0.00105)
       (* (cos re) (- im))
       (if (<= im 2e+103)
         t_0
         (if (<= im 2e+165)
           (* (* -0.3333333333333333 (pow im 3.0)) (+ 0.5 (* -0.25 (* re re))))
           (- (* (pow im 3.0) -0.16666666666666666) im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double tmp;
	if (im <= -0.0102) {
		tmp = t_0;
	} else if (im <= 0.00105) {
		tmp = cos(re) * -im;
	} else if (im <= 2e+103) {
		tmp = t_0;
	} else if (im <= 2e+165) {
		tmp = (-0.3333333333333333 * pow(im, 3.0)) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = (pow(im, 3.0) * -0.16666666666666666) - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    if (im <= (-0.0102d0)) then
        tmp = t_0
    else if (im <= 0.00105d0) then
        tmp = cos(re) * -im
    else if (im <= 2d+103) then
        tmp = t_0
    else if (im <= 2d+165) then
        tmp = ((-0.3333333333333333d0) * (im ** 3.0d0)) * (0.5d0 + ((-0.25d0) * (re * re)))
    else
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double tmp;
	if (im <= -0.0102) {
		tmp = t_0;
	} else if (im <= 0.00105) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 2e+103) {
		tmp = t_0;
	} else if (im <= 2e+165) {
		tmp = (-0.3333333333333333 * Math.pow(im, 3.0)) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	tmp = 0
	if im <= -0.0102:
		tmp = t_0
	elif im <= 0.00105:
		tmp = math.cos(re) * -im
	elif im <= 2e+103:
		tmp = t_0
	elif im <= 2e+165:
		tmp = (-0.3333333333333333 * math.pow(im, 3.0)) * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (im <= -0.0102)
		tmp = t_0;
	elseif (im <= 0.00105)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 2e+103)
		tmp = t_0;
	elseif (im <= 2e+165)
		tmp = Float64(Float64(-0.3333333333333333 * (im ^ 3.0)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	tmp = 0.0;
	if (im <= -0.0102)
		tmp = t_0;
	elseif (im <= 0.00105)
		tmp = cos(re) * -im;
	elseif (im <= 2e+103)
		tmp = t_0;
	elseif (im <= 2e+165)
		tmp = (-0.3333333333333333 * (im ^ 3.0)) * (0.5 + (-0.25 * (re * re)));
	else
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -0.0102], t$95$0, If[LessEqual[im, 0.00105], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 2e+103], t$95$0, If[LessEqual[im, 2e+165], N[(N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -0.010200000000000001 or 0.00104999999999999994 < im < 2e103

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 77.8%

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

   if -0.010200000000000001 < im < 0.00104999999999999994

   1. Initial program 9.4%

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

      1. neg-sub0 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in im around 0 98.7%

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

      1. associate-*r* 98.7%

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

      2. neg-mul-1 98.7%

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

   6. Simplified 98.7%

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

   if 2e103 < im < 1.9999999999999998e165

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 100.0%

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

      4. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 1.9999999999999998e165 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 65.4%

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

      4. unpow2 65.4%

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

   6. Simplified 65.4%

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

   7. Taylor expanded in im around 0 65.4%

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

   8. Taylor expanded in re around 0 96.2%

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

      1. distribute-lft-in 96.2%

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

      2. associate-*r* 96.2%

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

      3. metadata-eval 96.2%

         \[\leadsto \color{blue}{-1} \cdot im + 0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right) \]

      4. neg-mul-1 96.2%

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

      5. associate-*r* 96.2%

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

      6. metadata-eval 96.2%

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

      7. *-commutative 96.2%

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

      8. +-commutative 96.2%

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

      9. unsub-neg 96.2%

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

      10. *-commutative 96.2%

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

   10. Simplified 96.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.0102:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.00105:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+165}:\\ \;\;\;\;\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 8: 93.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -3.3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00059:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 5.0) (* (cos re) -0.008333333333333333))))
   (if (<= im -3.3)
     t_0
     (if (<= im 0.00059)
       (* (cos re) (- im))
       (if (<= im 1.6e+58) (* 0.5 (- (exp (- im)) (exp im))) t_0)))))

C

double code(double re, double im) {
	double t_0 = pow(im, 5.0) * (cos(re) * -0.008333333333333333);
	double tmp;
	if (im <= -3.3) {
		tmp = t_0;
	} else if (im <= 0.00059) {
		tmp = cos(re) * -im;
	} else if (im <= 1.6e+58) {
		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 ** 5.0d0) * (cos(re) * (-0.008333333333333333d0))
    if (im <= (-3.3d0)) then
        tmp = t_0
    else if (im <= 0.00059d0) then
        tmp = cos(re) * -im
    else if (im <= 1.6d+58) 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, 5.0) * (Math.cos(re) * -0.008333333333333333);
	double tmp;
	if (im <= -3.3) {
		tmp = t_0;
	} else if (im <= 0.00059) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.6e+58) {
		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, 5.0) * (math.cos(re) * -0.008333333333333333)
	tmp = 0
	if im <= -3.3:
		tmp = t_0
	elif im <= 0.00059:
		tmp = math.cos(re) * -im
	elif im <= 1.6e+58:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 5.0) * Float64(cos(re) * -0.008333333333333333))
	tmp = 0.0
	if (im <= -3.3)
		tmp = t_0;
	elseif (im <= 0.00059)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.6e+58)
		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 ^ 5.0) * (cos(re) * -0.008333333333333333);
	tmp = 0.0;
	if (im <= -3.3)
		tmp = t_0;
	elseif (im <= 0.00059)
		tmp = cos(re) * -im;
	elseif (im <= 1.6e+58)
		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, 5.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.3], t$95$0, If[LessEqual[im, 0.00059], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.6e+58], 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 < -3.2999999999999998 or 1.60000000000000008e58 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 92.5%

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

   5. Taylor expanded in im around inf 92.5%

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

      1. *-commutative 92.5%

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

      2. associate-*l* 92.5%

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

   7. Simplified 92.5%

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

   if -3.2999999999999998 < im < 5.9000000000000003e-4

   1. Initial program 9.4%

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

      1. neg-sub0 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in im around 0 98.7%

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

      1. associate-*r* 98.7%

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

      2. neg-mul-1 98.7%

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

   6. Simplified 98.7%

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

   if 5.9000000000000003e-4 < im < 1.60000000000000008e58

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 62.7%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.3:\\ \;\;\;\;{im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \mathbf{elif}\;im \leq 0.00059:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \end{array} \]

Alternative 9: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -4.2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00126:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 0.00126)
       (* (cos re) (- im))
       (if (<= im 1.1e+44) (* 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 <= 0.00126) {
		tmp = cos(re) * -im;
	} else if (im <= 1.1e+44) {
		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 <= 0.00126d0) then
        tmp = cos(re) * -im
    else if (im <= 1.1d+44) 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 <= 0.00126) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.1e+44) {
		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 <= 0.00126:
		tmp = math.cos(re) * -im
	elif im <= 1.1e+44:
		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 <= 0.00126)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.1e+44)
		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 <= 0.00126)
		tmp = cos(re) * -im;
	elseif (im <= 1.1e+44)
		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, 0.00126], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.1e+44], 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 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.3%

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

   5. Taylor expanded in im around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   7. Simplified 98.3%

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

   if -4.20000000000000018 < im < 0.00126000000000000005

   1. Initial program 9.4%

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

      1. neg-sub0 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in im around 0 98.7%

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

      1. associate-*r* 98.7%

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

      2. neg-mul-1 98.7%

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

   6. Simplified 98.7%

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

   if 0.00126000000000000005 < im < 1.09999999999999998e44

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 58.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.2:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq 0.00126:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 10: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -6.8:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00125:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 7.0) (* (cos re) -0.0001984126984126984))))
   (if (<= im -6.8)
     t_0
     (if (<= im 0.00125)
       (* (cos re) (- (* (pow im 5.0) -0.008333333333333333) im))
       (if (<= im 1.1e+44) (* 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 <= -6.8) {
		tmp = t_0;
	} else if (im <= 0.00125) {
		tmp = cos(re) * ((pow(im, 5.0) * -0.008333333333333333) - im);
	} else if (im <= 1.1e+44) {
		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 <= (-6.8d0)) then
        tmp = t_0
    else if (im <= 0.00125d0) then
        tmp = cos(re) * (((im ** 5.0d0) * (-0.008333333333333333d0)) - im)
    else if (im <= 1.1d+44) 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 <= -6.8) {
		tmp = t_0;
	} else if (im <= 0.00125) {
		tmp = Math.cos(re) * ((Math.pow(im, 5.0) * -0.008333333333333333) - im);
	} else if (im <= 1.1e+44) {
		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 <= -6.8:
		tmp = t_0
	elif im <= 0.00125:
		tmp = math.cos(re) * ((math.pow(im, 5.0) * -0.008333333333333333) - im)
	elif im <= 1.1e+44:
		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 <= -6.8)
		tmp = t_0;
	elseif (im <= 0.00125)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 5.0) * -0.008333333333333333) - im));
	elseif (im <= 1.1e+44)
		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 <= -6.8)
		tmp = t_0;
	elseif (im <= 0.00125)
		tmp = cos(re) * (((im ^ 5.0) * -0.008333333333333333) - im);
	elseif (im <= 1.1e+44)
		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, -6.8], t$95$0, If[LessEqual[im, 0.00125], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], 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 < -6.79999999999999982 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.3%

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

   5. Taylor expanded in im around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   7. Simplified 98.3%

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

   if -6.79999999999999982 < im < 0.00125000000000000003

   1. Initial program 9.4%

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

      1. neg-sub0 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in im around 0 99.8%

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

   5. Taylor expanded in im around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. neg-mul-1 99.8%

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

      3. unsub-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. associate-*r* 99.8%

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

      6. distribute-rgt-out 99.8%

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

      7. *-commutative 99.8%

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

      8. distribute-lft-out-- 99.8%

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

      9. *-commutative 99.8%

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

      10. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in im around inf 98.7%

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

   if 0.00125000000000000003 < im < 1.09999999999999998e44

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 58.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.8:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq 0.00125:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 11: 95.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -5.7:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.061:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 7.0) (* (cos re) -0.0001984126984126984))))
   (if (<= im -5.7)
     t_0
     (if (<= im 0.061)
       (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
       (if (<= im 1.1e+44) (* 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 <= -5.7) {
		tmp = t_0;
	} else if (im <= 0.061) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		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 <= (-5.7d0)) then
        tmp = t_0
    else if (im <= 0.061d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 1.1d+44) 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 <= -5.7) {
		tmp = t_0;
	} else if (im <= 0.061) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		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 <= -5.7:
		tmp = t_0
	elif im <= 0.061:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 1.1e+44:
		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 <= -5.7)
		tmp = t_0;
	elseif (im <= 0.061)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 1.1e+44)
		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 <= -5.7)
		tmp = t_0;
	elseif (im <= 0.061)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 1.1e+44)
		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, -5.7], t$95$0, If[LessEqual[im, 0.061], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.70000000000000018 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.3%

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

   5. Taylor expanded in im around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   7. Simplified 98.3%

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

   if -5.70000000000000018 < im < 0.060999999999999999

   1. Initial program 9.4%

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

      1. neg-sub0 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in im around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

      4. associate-*r* 99.5%

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

      5. distribute-rgt-out-- 99.5%

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

      6. *-commutative 99.5%

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

   6. Simplified 99.5%

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

   if 0.060999999999999999 < im < 1.09999999999999998e44

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 58.8%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.7:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq 0.061:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 12: 76.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -9.5 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 8.6 \cdot 10^{-8}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+165}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -9.5e+56)
     t_0
     (if (<= im 8.6e-8)
       (* (cos re) (- im))
       (if (<= im 8.5e+165)
         (*
          (+ 0.5 (* -0.25 (* re re)))
          (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -9.5e+56) {
		tmp = t_0;
	} else if (im <= 8.6e-8) {
		tmp = cos(re) * -im;
	} else if (im <= 8.5e+165) {
		tmp = (0.5 + (-0.25 * (re * re))) * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-9.5d+56)) then
        tmp = t_0
    else if (im <= 8.6d-8) then
        tmp = cos(re) * -im
    else if (im <= 8.5d+165) then
        tmp = (0.5d0 + ((-0.25d0) * (re * re))) * ((im * (-2.0d0)) + ((-0.3333333333333333d0) * (im ** 3.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -9.5e+56) {
		tmp = t_0;
	} else if (im <= 8.6e-8) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 8.5e+165) {
		tmp = (0.5 + (-0.25 * (re * re))) * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -9.5e+56:
		tmp = t_0
	elif im <= 8.6e-8:
		tmp = math.cos(re) * -im
	elif im <= 8.5e+165:
		tmp = (0.5 + (-0.25 * (re * re))) * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -9.5e+56)
		tmp = t_0;
	elseif (im <= 8.6e-8)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 8.5e+165)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * Float64(re * re))) * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -9.5e+56)
		tmp = t_0;
	elseif (im <= 8.6e-8)
		tmp = cos(re) * -im;
	elseif (im <= 8.5e+165)
		tmp = (0.5 + (-0.25 * (re * re))) * ((im * -2.0) + (-0.3333333333333333 * (im ^ 3.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -9.5e+56], t$95$0, If[LessEqual[im, 8.6e-8], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 8.5e+165], N[(N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.4999999999999997e56 or 8.5000000000000001e165 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 72.1%

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

      4. unpow2 72.1%

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

   6. Simplified 72.1%

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

   7. Taylor expanded in im around 0 62.3%

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

   8. Taylor expanded in re around 0 78.7%

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

      1. distribute-lft-in 78.7%

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

      2. associate-*r* 78.7%

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

      3. metadata-eval 78.7%

         \[\leadsto \color{blue}{-1} \cdot im + 0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right) \]

      4. neg-mul-1 78.7%

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

      5. associate-*r* 78.7%

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

      6. metadata-eval 78.7%

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

      7. *-commutative 78.7%

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

      8. +-commutative 78.7%

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

      9. unsub-neg 78.7%

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

      10. *-commutative 78.7%

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

   10. Simplified 78.7%

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

   if -9.4999999999999997e56 < im < 8.6000000000000002e-8

   1. Initial program 12.8%

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

      1. neg-sub0 12.8%

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

   3. Simplified 12.8%

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

   4. Taylor expanded in im around 0 93.7%

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

      1. associate-*r* 93.7%

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

      2. neg-mul-1 93.7%

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

   6. Simplified 93.7%

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

   if 8.6000000000000002e-8 < im < 8.5000000000000001e165

   1. Initial program 98.3%

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

      1. neg-sub0 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in re around 0 14.8%

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

      1. +-commutative 14.8%

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

      2. associate-*r* 14.8%

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

      3. distribute-rgt-out 86.9%

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

      4. unpow2 86.9%

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

   6. Simplified 86.9%

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

   7. Taylor expanded in im around 0 51.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9.5 \cdot 10^{+56}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 8.6 \cdot 10^{-8}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+165}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 13: 76.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -1.2 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 42000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 10^{+165}:\\ \;\;\;\;\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -1.2e+57)
     t_0
     (if (<= im 42000000000.0)
       (* (cos re) (- im))
       (if (<= im 1e+165)
         (* (* -0.3333333333333333 (pow im 3.0)) (+ 0.5 (* -0.25 (* re re))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -1.2e+57) {
		tmp = t_0;
	} else if (im <= 42000000000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 1e+165) {
		tmp = (-0.3333333333333333 * pow(im, 3.0)) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-1.2d+57)) then
        tmp = t_0
    else if (im <= 42000000000.0d0) then
        tmp = cos(re) * -im
    else if (im <= 1d+165) then
        tmp = ((-0.3333333333333333d0) * (im ** 3.0d0)) * (0.5d0 + ((-0.25d0) * (re * re)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -1.2e+57) {
		tmp = t_0;
	} else if (im <= 42000000000.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1e+165) {
		tmp = (-0.3333333333333333 * Math.pow(im, 3.0)) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -1.2e+57:
		tmp = t_0
	elif im <= 42000000000.0:
		tmp = math.cos(re) * -im
	elif im <= 1e+165:
		tmp = (-0.3333333333333333 * math.pow(im, 3.0)) * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -1.2e+57)
		tmp = t_0;
	elseif (im <= 42000000000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1e+165)
		tmp = Float64(Float64(-0.3333333333333333 * (im ^ 3.0)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -1.2e+57)
		tmp = t_0;
	elseif (im <= 42000000000.0)
		tmp = cos(re) * -im;
	elseif (im <= 1e+165)
		tmp = (-0.3333333333333333 * (im ^ 3.0)) * (0.5 + (-0.25 * (re * re)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -1.2e+57], t$95$0, If[LessEqual[im, 42000000000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1e+165], N[(N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.20000000000000002e57 or 9.99999999999999899e164 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 72.1%

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

      4. unpow2 72.1%

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

   6. Simplified 72.1%

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

   7. Taylor expanded in im around 0 62.3%

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

   8. Taylor expanded in re around 0 78.7%

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

      1. distribute-lft-in 78.7%

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

      2. associate-*r* 78.7%

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

      3. metadata-eval 78.7%

         \[\leadsto \color{blue}{-1} \cdot im + 0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right) \]

      4. neg-mul-1 78.7%

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

      5. associate-*r* 78.7%

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

      6. metadata-eval 78.7%

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

      7. *-commutative 78.7%

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

      8. +-commutative 78.7%

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

      9. unsub-neg 78.7%

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

      10. *-commutative 78.7%

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

   10. Simplified 78.7%

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

   if -1.20000000000000002e57 < im < 4.2e10

   1. Initial program 19.2%

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

      1. neg-sub0 19.2%

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

   3. Simplified 19.2%

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

   4. Taylor expanded in im around 0 88.7%

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

      1. associate-*r* 88.7%

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

      2. neg-mul-1 88.7%

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

   6. Simplified 88.7%

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

   if 4.2e10 < im < 9.99999999999999899e164

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 93.8%

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

      4. unpow2 93.8%

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

   6. Simplified 93.8%

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

   7. Taylor expanded in im around 0 58.1%

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

   8. Taylor expanded in im around inf 58.1%

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

      1. *-commutative 58.1%

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

   10. Simplified 58.1%

       \[\leadsto \color{blue}{\left({im}^{3} \cdot -0.3333333333333333\right)} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.2 \cdot 10^{+57}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 42000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 10^{+165}:\\ \;\;\;\;\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 14: 76.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -7.2 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 31000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 3.25 \cdot 10^{+103}:\\ \;\;\;\;{im}^{3} \cdot \left(\left(re \cdot re\right) \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -7.2e+56)
     t_0
     (if (<= im 31000000000.0)
       (* (cos re) (- im))
       (if (<= im 3.25e+103)
         (* (pow im 3.0) (* (* re re) 0.08333333333333333))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -7.2e+56) {
		tmp = t_0;
	} else if (im <= 31000000000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 3.25e+103) {
		tmp = pow(im, 3.0) * ((re * re) * 0.08333333333333333);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-7.2d+56)) then
        tmp = t_0
    else if (im <= 31000000000.0d0) then
        tmp = cos(re) * -im
    else if (im <= 3.25d+103) then
        tmp = (im ** 3.0d0) * ((re * re) * 0.08333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -7.2e+56) {
		tmp = t_0;
	} else if (im <= 31000000000.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 3.25e+103) {
		tmp = Math.pow(im, 3.0) * ((re * re) * 0.08333333333333333);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -7.2e+56:
		tmp = t_0
	elif im <= 31000000000.0:
		tmp = math.cos(re) * -im
	elif im <= 3.25e+103:
		tmp = math.pow(im, 3.0) * ((re * re) * 0.08333333333333333)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -7.2e+56)
		tmp = t_0;
	elseif (im <= 31000000000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 3.25e+103)
		tmp = Float64((im ^ 3.0) * Float64(Float64(re * re) * 0.08333333333333333));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -7.2e+56)
		tmp = t_0;
	elseif (im <= 31000000000.0)
		tmp = cos(re) * -im;
	elseif (im <= 3.25e+103)
		tmp = (im ^ 3.0) * ((re * re) * 0.08333333333333333);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -7.2e+56], t$95$0, If[LessEqual[im, 31000000000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 3.25e+103], N[(N[Power[im, 3.0], $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.19999999999999996e56 or 3.25000000000000001e103 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 75.5%

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

      4. unpow2 75.5%

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

   6. Simplified 75.5%

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

   7. Taylor expanded in im around 0 66.9%

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

   8. Taylor expanded in re around 0 75.2%

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

      1. distribute-lft-in 75.2%

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

      2. associate-*r* 75.2%

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

      3. metadata-eval 75.2%

         \[\leadsto \color{blue}{-1} \cdot im + 0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right) \]

      4. neg-mul-1 75.2%

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

      5. associate-*r* 75.2%

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

      6. metadata-eval 75.2%

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

      7. *-commutative 75.2%

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

      8. +-commutative 75.2%

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

      9. unsub-neg 75.2%

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

      10. *-commutative 75.2%

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

   10. Simplified 75.2%

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

   if -7.19999999999999996e56 < im < 3.1e10

   1. Initial program 19.2%

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

      1. neg-sub0 19.2%

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

   3. Simplified 19.2%

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

   4. Taylor expanded in im around 0 88.7%

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

      1. associate-*r* 88.7%

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

      2. neg-mul-1 88.7%

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

   6. Simplified 88.7%

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

   if 3.1e10 < im < 3.25000000000000001e103

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 90.0%

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

      4. unpow2 90.0%

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

   6. Simplified 90.0%

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

   7. Taylor expanded in im around 0 33.0%

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

   8. Taylor expanded in im around inf 33.0%

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

      1. *-commutative 33.0%

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

   10. Simplified 33.0%

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

   11. Taylor expanded in re around inf 31.3%

       \[\leadsto \color{blue}{0.08333333333333333 \cdot \left({im}^{3} \cdot {re}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 31.3%

          \[\leadsto \color{blue}{\left({im}^{3} \cdot {re}^{2}\right) \cdot 0.08333333333333333} \]

       2. associate-*l* 31.3%

          \[\leadsto \color{blue}{{im}^{3} \cdot \left({re}^{2} \cdot 0.08333333333333333\right)} \]

       3. unpow2 31.3%

          \[\leadsto {im}^{3} \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot 0.08333333333333333\right) \]

   13. Simplified 31.3%

       \[\leadsto \color{blue}{{im}^{3} \cdot \left(\left(re \cdot re\right) \cdot 0.08333333333333333\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{+56}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 31000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 3.25 \cdot 10^{+103}:\\ \;\;\;\;{im}^{3} \cdot \left(\left(re \cdot re\right) \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 15: 75.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -9.5 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 35000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -9.5e+56)
     t_0
     (if (<= im 35000000000.0)
       (* (cos re) (- im))
       (if (<= im 4.2e+70) (- (* 0.5 (* im (* re re))) im) t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -9.5e+56) {
		tmp = t_0;
	} else if (im <= 35000000000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 4.2e+70) {
		tmp = (0.5 * (im * (re * re))) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-9.5d+56)) then
        tmp = t_0
    else if (im <= 35000000000.0d0) then
        tmp = cos(re) * -im
    else if (im <= 4.2d+70) then
        tmp = (0.5d0 * (im * (re * re))) - im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -9.5e+56) {
		tmp = t_0;
	} else if (im <= 35000000000.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 4.2e+70) {
		tmp = (0.5 * (im * (re * re))) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -9.5e+56:
		tmp = t_0
	elif im <= 35000000000.0:
		tmp = math.cos(re) * -im
	elif im <= 4.2e+70:
		tmp = (0.5 * (im * (re * re))) - im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -9.5e+56)
		tmp = t_0;
	elseif (im <= 35000000000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 4.2e+70)
		tmp = Float64(Float64(0.5 * Float64(im * Float64(re * re))) - im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -9.5e+56)
		tmp = t_0;
	elseif (im <= 35000000000.0)
		tmp = cos(re) * -im;
	elseif (im <= 4.2e+70)
		tmp = (0.5 * (im * (re * re))) - im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -9.5e+56], t$95$0, If[LessEqual[im, 35000000000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 4.2e+70], N[(N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.4999999999999997e56 or 4.20000000000000015e70 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 76.5%

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

      4. unpow2 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in im around 0 65.5%

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

   8. Taylor expanded in re around 0 72.4%

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

      1. distribute-lft-in 72.4%

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

      2. associate-*r* 72.4%

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

      3. metadata-eval 72.4%

         \[\leadsto \color{blue}{-1} \cdot im + 0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right) \]

      4. neg-mul-1 72.4%

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

      5. associate-*r* 72.4%

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

      6. metadata-eval 72.4%

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

      7. *-commutative 72.4%

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

      8. +-commutative 72.4%

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

      9. unsub-neg 72.4%

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

      10. *-commutative 72.4%

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

   10. Simplified 72.4%

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

   if -9.4999999999999997e56 < im < 3.5e10

   1. Initial program 19.2%

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

      1. neg-sub0 19.2%

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

   3. Simplified 19.2%

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

   4. Taylor expanded in im around 0 88.7%

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

      1. associate-*r* 88.7%

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

      2. neg-mul-1 88.7%

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

   6. Simplified 88.7%

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

   if 3.5e10 < im < 4.20000000000000015e70

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.4%

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

      1. associate-*r* 3.4%

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

      2. neg-mul-1 3.4%

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

   6. Simplified 3.4%

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

   7. Taylor expanded in re around 0 33.3%

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

      1. neg-mul-1 33.3%

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

      2. +-commutative 33.3%

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

      3. unsub-neg 33.3%

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

      4. unpow2 33.3%

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

   9. Simplified 33.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9.5 \cdot 10^{+56}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 35000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 16: 59.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{if}\;im \leq -1.25 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 100000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+182} \lor \neg \left(im \leq 5.5 \cdot 10^{+272}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot -6.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (* im (* re re))) im)))
   (if (<= im -1.25e+50)
     t_0
     (if (<= im 100000000000.0)
       (* (cos re) (- im))
       (if (or (<= im 2.4e+182) (not (<= im 5.5e+272)))
         t_0
         (* re (* re -6.75)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * (im * (re * re))) - im;
	double tmp;
	if (im <= -1.25e+50) {
		tmp = t_0;
	} else if (im <= 100000000000.0) {
		tmp = cos(re) * -im;
	} else if ((im <= 2.4e+182) || !(im <= 5.5e+272)) {
		tmp = t_0;
	} else {
		tmp = re * (re * -6.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * (im * (re * re))) - im
    if (im <= (-1.25d+50)) then
        tmp = t_0
    else if (im <= 100000000000.0d0) then
        tmp = cos(re) * -im
    else if ((im <= 2.4d+182) .or. (.not. (im <= 5.5d+272))) then
        tmp = t_0
    else
        tmp = re * (re * (-6.75d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * (im * (re * re))) - im;
	double tmp;
	if (im <= -1.25e+50) {
		tmp = t_0;
	} else if (im <= 100000000000.0) {
		tmp = Math.cos(re) * -im;
	} else if ((im <= 2.4e+182) || !(im <= 5.5e+272)) {
		tmp = t_0;
	} else {
		tmp = re * (re * -6.75);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * (im * (re * re))) - im
	tmp = 0
	if im <= -1.25e+50:
		tmp = t_0
	elif im <= 100000000000.0:
		tmp = math.cos(re) * -im
	elif (im <= 2.4e+182) or not (im <= 5.5e+272):
		tmp = t_0
	else:
		tmp = re * (re * -6.75)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * Float64(im * Float64(re * re))) - im)
	tmp = 0.0
	if (im <= -1.25e+50)
		tmp = t_0;
	elseif (im <= 100000000000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif ((im <= 2.4e+182) || !(im <= 5.5e+272))
		tmp = t_0;
	else
		tmp = Float64(re * Float64(re * -6.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * (im * (re * re))) - im;
	tmp = 0.0;
	if (im <= -1.25e+50)
		tmp = t_0;
	elseif (im <= 100000000000.0)
		tmp = cos(re) * -im;
	elseif ((im <= 2.4e+182) || ~((im <= 5.5e+272)))
		tmp = t_0;
	else
		tmp = re * (re * -6.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -1.25e+50], t$95$0, If[LessEqual[im, 100000000000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[Or[LessEqual[im, 2.4e+182], N[Not[LessEqual[im, 5.5e+272]], $MachinePrecision]], t$95$0, N[(re * N[(re * -6.75), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.25e50 or 1e11 < im < 2.4000000000000001e182 or 5.4999999999999998e272 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.3%

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

      1. associate-*r* 5.3%

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

      2. neg-mul-1 5.3%

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

   6. Simplified 5.3%

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

   7. Taylor expanded in re around 0 21.6%

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

      1. neg-mul-1 21.6%

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

      2. +-commutative 21.6%

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

      3. unsub-neg 21.6%

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

      4. unpow2 21.6%

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

   9. Simplified 21.6%

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

   if -1.25e50 < im < 1e11

   1. Initial program 17.4%

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

      1. neg-sub0 17.4%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in im around 0 90.6%

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

      1. associate-*r* 90.6%

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

      2. neg-mul-1 90.6%

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

   6. Simplified 90.6%

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

   if 2.4000000000000001e182 < im < 5.4999999999999998e272

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 52.9%

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

      4. unpow2 52.9%

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

   6. Simplified 52.9%

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

   8. Applied egg-rr 30.3%

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

   9. Taylor expanded in re around inf 31.0%

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

       1. *-commutative 31.0%

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

       2. unpow2 31.0%

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

       3. associate-*l* 31.0%

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

   11. Simplified 31.0%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.25 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 100000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+182} \lor \neg \left(im \leq 5.5 \cdot 10^{+272}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot -6.75\right)\\ \end{array} \]

Alternative 17: 35.6% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

   1. neg-sub0 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 50.3%

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

   1. associate-*r* 50.3%

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

   2. neg-mul-1 50.3%

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

6. Simplified 50.3%

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

7. Taylor expanded in re around 0 36.0%

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

   1. neg-mul-1 36.0%

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

   2. +-commutative 36.0%

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

   3. unsub-neg 36.0%

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

   4. unpow2 36.0%

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

9. Simplified 36.0%

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

10. Final simplification 36.0%

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

Alternative 18: 31.0% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+170}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot -6.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5e+170) (* 0.5 (* im -2.0)) (* re (* re -6.75))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5e+170) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = re * (re * -6.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5d+170) then
        tmp = 0.5d0 * (im * (-2.0d0))
    else
        tmp = re * (re * (-6.75d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5e+170) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = re * (re * -6.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5e+170:
		tmp = 0.5 * (im * -2.0)
	else:
		tmp = re * (re * -6.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5e+170)
		tmp = Float64(0.5 * Float64(im * -2.0));
	else
		tmp = Float64(re * Float64(re * -6.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5e+170)
		tmp = 0.5 * (im * -2.0);
	else
		tmp = re * (re * -6.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5e+170], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * -6.75), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.99999999999999977e170

   1. Initial program 55.4%

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

      1. neg-sub0 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in re around 0 45.4%

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

   5. Taylor expanded in im around 0 34.4%

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

   if 4.99999999999999977e170 < re

   1. Initial program 64.9%

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

      1. neg-sub0 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 27.6%

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

      4. unpow2 27.6%

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

   6. Simplified 27.6%

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

   8. Applied egg-rr 45.4%

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

   9. Taylor expanded in re around inf 45.4%

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

       1. *-commutative 45.4%

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

       2. unpow2 45.4%

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

       3. associate-*l* 45.4%

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

   11. Simplified 45.4%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+170}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot -6.75\right)\\ \end{array} \]

Alternative 19: 29.2% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot -2\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* im -2.0)))

C

double code(double re, double im) {
	return 0.5 * (im * -2.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (im * (-2.0d0))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (im * -2.0);
}

Python

def code(re, im):
	return 0.5 * (im * -2.0)

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * -2.0))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im * -2.0);
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

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. neg-sub0 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 re around 0 44.5%

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

5. Taylor expanded in im around 0 31.1%

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

6. Final simplification 31.1%

   \[\leadsto 0.5 \cdot \left(im \cdot -2\right) \]

Alternative 20: 29.1% 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 56.5%

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

   1. neg-sub0 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 50.3%

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

   1. associate-*r* 50.3%

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

   2. neg-mul-1 50.3%

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

6. Simplified 50.3%

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

7. Taylor expanded in re around 0 30.7%

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

   1. neg-mul-1 30.7%

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

9. Simplified 30.7%

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

10. Final simplification 30.7%

    \[\leadsto -im \]

Alternative 21: 2.9% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 13.5)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 13.5

Julia

function code(re, im)
	return 13.5
end

MATLAB

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

Wolfram

code[re_, im_] := 13.5

Derivation

1. Initial program 56.5%

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

   1. neg-sub0 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 re around 0 4.6%

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

   1. +-commutative 4.6%

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

   2. associate-*r* 4.6%

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

   3. distribute-rgt-out 42.5%

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

   4. unpow2 42.5%

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

6. Simplified 42.5%

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

8. Applied egg-rr 8.4%

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

9. Taylor expanded in re around 0 2.9%

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

10. Final simplification 2.9%

    \[\leadsto 13.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023293 
(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))))