math.sin on complex, imaginary part

Percentage Accurate: 53.6% → 99.6%

Time: 14.8s

Alternatives: 14

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: 53.6% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -5 \lor \neg \left(t_0 \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 + {im}^{5} \cdot -0.008333333333333333\right) - im \cdot \cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -5.0) (not (<= t_0 2e-12)))
     (* (* 0.5 (cos re)) t_0)
     (-
      (*
       (cos re)
       (+
        (* (pow im 3.0) -0.16666666666666666)
        (* (pow im 5.0) -0.008333333333333333)))
      (* im (cos re))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 2e-12)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = (cos(re) * ((pow(im, 3.0) * -0.16666666666666666) + (pow(im, 5.0) * -0.008333333333333333))) - (im * cos(re));
	}
	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 <= (-5.0d0)) .or. (.not. (t_0 <= 2d-12))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = (cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) + ((im ** 5.0d0) * (-0.008333333333333333d0)))) - (im * cos(re))
    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 <= -5.0) || !(t_0 <= 2e-12)) {
		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 * Math.cos(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -5.0) or not (t_0 <= 2e-12):
		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 * math.cos(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -5.0) || !(t_0 <= 2e-12))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) + Float64((im ^ 5.0) * -0.008333333333333333))) - Float64(im * cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -5.0) || ~((t_0 <= 2e-12)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = (cos(re) * (((im ^ 3.0) * -0.16666666666666666) + ((im ^ 5.0) * -0.008333333333333333))) - (im * cos(re));
	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, -5.0], N[Not[LessEqual[t$95$0, 2e-12]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -5 or 1.99999999999999996e-12 < (-.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 -5 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1.99999999999999996e-12

   1. Initial program 9.3%

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

      1. neg-sub0 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-*r* 99.9%

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

      6. associate-*r* 99.9%

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

      7. distribute-rgt-out 99.9%

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

      8. +-commutative 99.9%

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

      9. *-commutative 99.9%

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

      10. *-commutative 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 + {im}^{5} \cdot -0.008333333333333333\right) - im \cdot \cos re\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \cos re\\ \mathbf{if}\;t_0 \leq -5 \lor \neg \left(t_0 \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left({im}^{3} \cdot -0.3333333333333333 + {im}^{5} \cdot -0.016666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (cos re))))
   (if (or (<= t_0 -5.0) (not (<= t_0 2e-12)))
     (* t_1 t_0)
     (*
      t_1
      (+
       (* im -2.0)
       (+
        (* (pow im 3.0) -0.3333333333333333)
        (* (pow im 5.0) -0.016666666666666666)))))))

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 <= -5.0) || !(t_0 <= 2e-12)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_1 * ((im * -2.0) + ((pow(im, 3.0) * -0.3333333333333333) + (pow(im, 5.0) * -0.016666666666666666)));
	}
	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 <= (-5.0d0)) .or. (.not. (t_0 <= 2d-12))) then
        tmp = t_1 * t_0
    else
        tmp = t_1 * ((im * (-2.0d0)) + (((im ** 3.0d0) * (-0.3333333333333333d0)) + ((im ** 5.0d0) * (-0.016666666666666666d0))))
    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 <= -5.0) || !(t_0 <= 2e-12)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_1 * ((im * -2.0) + ((Math.pow(im, 3.0) * -0.3333333333333333) + (Math.pow(im, 5.0) * -0.016666666666666666)));
	}
	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 <= -5.0) or not (t_0 <= 2e-12):
		tmp = t_1 * t_0
	else:
		tmp = t_1 * ((im * -2.0) + ((math.pow(im, 3.0) * -0.3333333333333333) + (math.pow(im, 5.0) * -0.016666666666666666)))
	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 <= -5.0) || !(t_0 <= 2e-12))
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(t_1 * Float64(Float64(im * -2.0) + Float64(Float64((im ^ 3.0) * -0.3333333333333333) + Float64((im ^ 5.0) * -0.016666666666666666))));
	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 <= -5.0) || ~((t_0 <= 2e-12)))
		tmp = t_1 * t_0;
	else
		tmp = t_1 * ((im * -2.0) + (((im ^ 3.0) * -0.3333333333333333) + ((im ^ 5.0) * -0.016666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5.0], N[Not[LessEqual[t$95$0, 2e-12]], $MachinePrecision]], N[(t$95$1 * t$95$0), $MachinePrecision], N[(t$95$1 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -5 or 1.99999999999999996e-12 < (-.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 -5 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1.99999999999999996e-12

   1. Initial program 9.3%

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

      1. neg-sub0 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in im around 0 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot -2 + \left({im}^{3} \cdot -0.3333333333333333 + {im}^{5} \cdot -0.016666666666666666\right)\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.002 \lor \neg \left(t_0 \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.002) (not (<= t_0 2e-12)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.002) || !(t_0 <= 2e-12)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.002d0)) .or. (.not. (t_0 <= 2d-12))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -0.002) || !(t_0 <= 2e-12)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.002) or not (t_0 <= 2e-12):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.002) || !(t_0 <= 2e-12))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.002) || ~((t_0 <= 2e-12)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.002], N[Not[LessEqual[t$95$0, 2e-12]], $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)) < -2e-3 or 1.99999999999999996e-12 < (-.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 -2e-3 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1.99999999999999996e-12

   1. Initial program 8.7%

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

      1. neg-sub0 8.7%

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

   3. Simplified 8.7%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out-- 99.9%

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

      6. *-commutative 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.002 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 4: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+94} \lor \neg \left(im \leq -130 \lor \neg \left(im \leq 2\right) \land im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -7.5e+94)
         (not (or (<= im -130.0) (and (not (<= im 2.0)) (<= im 5.6e+102)))))
   (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
   (* (- (exp (- im)) (exp im)) (+ 0.5 (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -7.5e+94) || !((im <= -130.0) || (!(im <= 2.0) && (im <= 5.6e+102)))) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = (exp(-im) - exp(im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-7.5d+94)) .or. (.not. (im <= (-130.0d0)) .or. (.not. (im <= 2.0d0)) .and. (im <= 5.6d+102))) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = (exp(-im) - exp(im)) * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -7.5e+94) || !((im <= -130.0) || (!(im <= 2.0) && (im <= 5.6e+102)))) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = (Math.exp(-im) - Math.exp(im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -7.5e+94) or not ((im <= -130.0) or (not (im <= 2.0) and (im <= 5.6e+102))):
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = (math.exp(-im) - math.exp(im)) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -7.5e+94) || !((im <= -130.0) || (!(im <= 2.0) && (im <= 5.6e+102))))
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -7.5e+94) || ~(((im <= -130.0) || (~((im <= 2.0)) && (im <= 5.6e+102)))))
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = (exp(-im) - exp(im)) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -7.5e+94], N[Not[Or[LessEqual[im, -130.0], And[N[Not[LessEqual[im, 2.0]], $MachinePrecision], LessEqual[im, 5.6e+102]]]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -7.49999999999999978e94 or -130 < im < 2 or 5.60000000000000037e102 < im

   1. Initial program 46.3%

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

      1. neg-sub0 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in im around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. mul-1-neg 98.3%

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

      3. unsub-neg 98.3%

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

      4. associate-*r* 98.3%

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

      5. distribute-rgt-out-- 98.3%

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

      6. *-commutative 98.3%

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

   6. Simplified 98.3%

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

   if -7.49999999999999978e94 < im < -130 or 2 < im < 5.60000000000000037e102

   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 3.0%

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

      1. +-commutative 3.0%

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

      2. associate-*r* 3.0%

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

      3. distribute-rgt-out 84.8%

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

      4. unpow2 84.8%

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

   6. Simplified 84.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+94} \lor \neg \left(im \leq -130 \lor \neg \left(im \leq 2\right) \land im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 5: 95.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{+102} \lor \neg \left(im \leq -0.072 \lor \neg \left(im \leq 2\right) \land im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -5.5e+102)
         (not (or (<= im -0.072) (and (not (<= im 2.0)) (<= im 5.6e+102)))))
   (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
   (* 0.5 (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -5.5e+102) || !((im <= -0.072) || (!(im <= 2.0) && (im <= 5.6e+102)))) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-5.5d+102)) .or. (.not. (im <= (-0.072d0)) .or. (.not. (im <= 2.0d0)) .and. (im <= 5.6d+102))) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = 0.5d0 * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -5.5e+102) || !((im <= -0.072) || (!(im <= 2.0) && (im <= 5.6e+102)))) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -5.5e+102) or not ((im <= -0.072) or (not (im <= 2.0) and (im <= 5.6e+102))):
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -5.5e+102) || !((im <= -0.072) || (!(im <= 2.0) && (im <= 5.6e+102))))
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -5.5e+102) || ~(((im <= -0.072) || (~((im <= 2.0)) && (im <= 5.6e+102)))))
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = 0.5 * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -5.5e+102], N[Not[Or[LessEqual[im, -0.072], And[N[Not[LessEqual[im, 2.0]], $MachinePrecision], LessEqual[im, 5.6e+102]]]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -5.49999999999999981e102 or -0.0719999999999999946 < im < 2 or 5.60000000000000037e102 < im

   1. Initial program 45.6%

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

      1. neg-sub0 45.6%

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

   3. Simplified 45.6%

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

   4. Taylor expanded in im around 0 99.5%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   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 -5.49999999999999981e102 < im < -0.0719999999999999946 or 2 < im < 5.60000000000000037e102

   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 70.0%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{+102} \lor \neg \left(im \leq -0.072 \lor \neg \left(im \leq 2\right) \land im \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 6: 85.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -3 \cdot 10^{+234}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.1 \cdot 10^{+70}:\\ \;\;\;\;t_0 \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{elif}\;im \leq -0.00105 \lor \neg \left(im \leq 2\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -3e+234)
     t_0
     (if (<= im -2.1e+70)
       (* t_0 (+ 1.0 (* re (* re -0.5))))
       (if (or (<= im -0.00105) (not (<= im 2.0)))
         (* 0.5 (- (exp (- im)) (exp im)))
         (* im (- (cos re))))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -3e+234) {
		tmp = t_0;
	} else if (im <= -2.1e+70) {
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	} else if ((im <= -0.00105) || !(im <= 2.0)) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else {
		tmp = im * -cos(re);
	}
	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 <= (-3d+234)) then
        tmp = t_0
    else if (im <= (-2.1d+70)) then
        tmp = t_0 * (1.0d0 + (re * (re * (-0.5d0))))
    else if ((im <= (-0.00105d0)) .or. (.not. (im <= 2.0d0))) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else
        tmp = im * -cos(re)
    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 <= -3e+234) {
		tmp = t_0;
	} else if (im <= -2.1e+70) {
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	} else if ((im <= -0.00105) || !(im <= 2.0)) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -3e+234:
		tmp = t_0
	elif im <= -2.1e+70:
		tmp = t_0 * (1.0 + (re * (re * -0.5)))
	elif (im <= -0.00105) or not (im <= 2.0):
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -3e+234)
		tmp = t_0;
	elseif (im <= -2.1e+70)
		tmp = Float64(t_0 * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	elseif ((im <= -0.00105) || !(im <= 2.0))
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -3e+234)
		tmp = t_0;
	elseif (im <= -2.1e+70)
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	elseif ((im <= -0.00105) || ~((im <= 2.0)))
		tmp = 0.5 * (exp(-im) - exp(im));
	else
		tmp = im * -cos(re);
	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, -3e+234], t$95$0, If[LessEqual[im, -2.1e+70], N[(t$95$0 * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -0.00105], N[Not[LessEqual[im, 2.0]], $MachinePrecision]], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.9999999999999999e234

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 90.0%

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

   if -2.9999999999999999e234 < im < -2.10000000000000008e70

   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 83.6%

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

      1. +-commutative 83.6%

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

      2. mul-1-neg 83.6%

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

      3. unsub-neg 83.6%

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

      4. associate-*r* 83.6%

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

      5. distribute-rgt-out-- 83.6%

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

      6. *-commutative 83.6%

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

   6. Simplified 83.6%

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

   7. Taylor expanded in re around 0 9.0%

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

      1. associate--l+ 9.0%

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

      2. associate-*r* 9.0%

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

      3. distribute-lft1-in 76.6%

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

      4. *-commutative 76.6%

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

      5. +-commutative 76.6%

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

      6. unpow2 76.6%

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

      7. associate-*r* 76.6%

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

      8. *-commutative 76.6%

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

   9. Simplified 76.6%

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

   if -2.10000000000000008e70 < im < -0.00104999999999999994 or 2 < 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 72.7%

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

   if -0.00104999999999999994 < im < 2

   1. Initial program 10.0%

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

      1. neg-sub0 10.0%

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

   3. Simplified 10.0%

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

   4. Taylor expanded in im around 0 98.5%

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

      1. associate-*r* 98.5%

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

      2. neg-mul-1 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3 \cdot 10^{+234}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq -2.1 \cdot 10^{+70}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{elif}\;im \leq -0.00105 \lor \neg \left(im \leq 2\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 7: 77.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -7.8 \cdot 10^{+233}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -49000000 \lor \neg \left(im \leq 6.5\right):\\ \;\;\;\;t_0 \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -7.8e+233)
     t_0
     (if (or (<= im -49000000.0) (not (<= im 6.5)))
       (* t_0 (+ 1.0 (* re (* re -0.5))))
       (* im (- (cos re)))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -7.8e+233) {
		tmp = t_0;
	} else if ((im <= -49000000.0) || !(im <= 6.5)) {
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = im * -cos(re);
	}
	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 <= (-7.8d+233)) then
        tmp = t_0
    else if ((im <= (-49000000.0d0)) .or. (.not. (im <= 6.5d0))) then
        tmp = t_0 * (1.0d0 + (re * (re * (-0.5d0))))
    else
        tmp = im * -cos(re)
    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 <= -7.8e+233) {
		tmp = t_0;
	} else if ((im <= -49000000.0) || !(im <= 6.5)) {
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -7.8e+233:
		tmp = t_0
	elif (im <= -49000000.0) or not (im <= 6.5):
		tmp = t_0 * (1.0 + (re * (re * -0.5)))
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -7.8e+233)
		tmp = t_0;
	elseif ((im <= -49000000.0) || !(im <= 6.5))
		tmp = Float64(t_0 * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -7.8e+233)
		tmp = t_0;
	elseif ((im <= -49000000.0) || ~((im <= 6.5)))
		tmp = t_0 * (1.0 + (re * (re * -0.5)));
	else
		tmp = im * -cos(re);
	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, -7.8e+233], t$95$0, If[Or[LessEqual[im, -49000000.0], N[Not[LessEqual[im, 6.5]], $MachinePrecision]], N[(t$95$0 * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.7999999999999998e233

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 90.0%

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

   if -7.7999999999999998e233 < im < -4.9e7 or 6.5 < im

   1. Initial program 100.0%

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

      1. 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 70.2%

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

      1. +-commutative 70.2%

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

      2. mul-1-neg 70.2%

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

      3. unsub-neg 70.2%

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

      4. associate-*r* 70.2%

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

      5. distribute-rgt-out-- 70.2%

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

      6. *-commutative 70.2%

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

   6. Simplified 70.2%

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

   7. Taylor expanded in re around 0 10.2%

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

      1. associate--l+ 10.2%

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

      2. associate-*r* 10.2%

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

      3. distribute-lft1-in 60.2%

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

      4. *-commutative 60.2%

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

      5. +-commutative 60.2%

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

      6. unpow2 60.2%

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

      7. associate-*r* 60.2%

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

      8. *-commutative 60.2%

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

   9. Simplified 60.2%

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

   if -4.9e7 < im < 6.5

   1. Initial program 13.3%

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

      1. neg-sub0 13.3%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in im around 0 95.1%

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

      1. associate-*r* 95.1%

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

      2. neg-mul-1 95.1%

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

   6. Simplified 95.1%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.8 \cdot 10^{+233}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq -49000000 \lor \neg \left(im \leq 6.5\right):\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 8: 75.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -2.75 \cdot 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6.8 \cdot 10^{+16}:\\ \;\;\;\;re \cdot \left(t_0 \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{elif}\;im \leq 2.35 \cdot 10^{+53}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -2.75e+122)
     t_0
     (if (<= im -6.8e+16)
       (* re (* t_0 (* re -0.5)))
       (if (<= im 2.35e+53) (* im (- (cos re))) t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -2.75e+122) {
		tmp = t_0;
	} else if (im <= -6.8e+16) {
		tmp = re * (t_0 * (re * -0.5));
	} else if (im <= 2.35e+53) {
		tmp = im * -cos(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 <= (-2.75d+122)) then
        tmp = t_0
    else if (im <= (-6.8d+16)) then
        tmp = re * (t_0 * (re * (-0.5d0)))
    else if (im <= 2.35d+53) then
        tmp = im * -cos(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 <= -2.75e+122) {
		tmp = t_0;
	} else if (im <= -6.8e+16) {
		tmp = re * (t_0 * (re * -0.5));
	} else if (im <= 2.35e+53) {
		tmp = im * -Math.cos(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 <= -2.75e+122:
		tmp = t_0
	elif im <= -6.8e+16:
		tmp = re * (t_0 * (re * -0.5))
	elif im <= 2.35e+53:
		tmp = im * -math.cos(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 <= -2.75e+122)
		tmp = t_0;
	elseif (im <= -6.8e+16)
		tmp = Float64(re * Float64(t_0 * Float64(re * -0.5)));
	elseif (im <= 2.35e+53)
		tmp = Float64(im * Float64(-cos(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 <= -2.75e+122)
		tmp = t_0;
	elseif (im <= -6.8e+16)
		tmp = re * (t_0 * (re * -0.5));
	elseif (im <= 2.35e+53)
		tmp = im * -cos(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, -2.75e+122], t$95$0, If[LessEqual[im, -6.8e+16], N[(re * N[(t$95$0 * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.35e+53], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.7499999999999999e122 or 2.34999999999999988e53 < 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 91.9%

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

      1. +-commutative 91.9%

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

      2. mul-1-neg 91.9%

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

      3. unsub-neg 91.9%

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

      4. associate-*r* 91.9%

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

      5. distribute-rgt-out-- 91.9%

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

      6. *-commutative 91.9%

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

   6. Simplified 91.9%

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

   7. Taylor expanded in re around 0 69.0%

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

   if -2.7499999999999999e122 < im < -6.8e16

   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 21.3%

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

      1. +-commutative 21.3%

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

      2. mul-1-neg 21.3%

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

      3. unsub-neg 21.3%

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

      4. associate-*r* 21.3%

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

      5. distribute-rgt-out-- 21.3%

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

      6. *-commutative 21.3%

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

   6. Simplified 21.3%

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

   7. Taylor expanded in re around 0 29.3%

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

      1. associate--l+ 29.3%

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

      2. associate-*r* 29.3%

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

      3. distribute-lft1-in 46.0%

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

      4. *-commutative 46.0%

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

      5. +-commutative 46.0%

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

      6. unpow2 46.0%

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

      7. associate-*r* 46.0%

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

      8. *-commutative 46.0%

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

   9. Simplified 46.0%

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

   10. Taylor expanded in re around inf 39.3%

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

       1. associate-*r* 39.3%

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

       2. *-commutative 39.3%

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

       3. unpow2 39.3%

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

       4. associate-*r* 39.3%

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

       5. *-commutative 39.3%

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

       6. associate-*l* 39.3%

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

       7. *-commutative 39.3%

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

   12. Simplified 39.3%

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

   if -6.8e16 < im < 2.34999999999999988e53

   1. Initial program 18.0%

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

      1. neg-sub0 18.0%

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

   3. Simplified 18.0%

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

   4. Taylor expanded in im around 0 90.1%

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

      1. associate-*r* 90.1%

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

      2. neg-mul-1 90.1%

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

   6. Simplified 90.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.75 \cdot 10^{+122}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq -6.8 \cdot 10^{+16}:\\ \;\;\;\;re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{elif}\;im \leq 2.35 \cdot 10^{+53}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 9: 75.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+86} \lor \neg \left(im \leq 1.7 \cdot 10^{+53}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -6.5e+86) (not (<= im 1.7e+53)))
   (- (* (pow im 3.0) -0.16666666666666666) im)
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -6.5e+86) || !(im <= 1.7e+53)) {
		tmp = (pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-6.5d+86)) .or. (.not. (im <= 1.7d+53))) then
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -6.5e+86) || !(im <= 1.7e+53)) {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -6.5e+86) or not (im <= 1.7e+53):
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -6.5e+86) || !(im <= 1.7e+53))
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -6.5e+86) || ~((im <= 1.7e+53)))
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -6.5e+86], N[Not[LessEqual[im, 1.7e+53]], $MachinePrecision]], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -6.49999999999999996e86 or 1.69999999999999999e53 < 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 90.4%

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

      1. +-commutative 90.4%

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

      2. mul-1-neg 90.4%

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

      3. unsub-neg 90.4%

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

      4. associate-*r* 90.4%

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

      5. distribute-rgt-out-- 90.4%

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

      6. *-commutative 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in re around 0 66.7%

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

   if -6.49999999999999996e86 < im < 1.69999999999999999e53

   1. Initial program 24.7%

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

      1. neg-sub0 24.7%

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

   3. Simplified 24.7%

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

   4. Taylor expanded in im around 0 83.0%

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

      1. associate-*r* 83.0%

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

      2. neg-mul-1 83.0%

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

   6. Simplified 83.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+86} \lor \neg \left(im \leq 1.7 \cdot 10^{+53}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 10: 60.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -8.5 \cdot 10^{+16} \lor \neg \left(im \leq 6.5\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -8.5e+16) (not (<= im 6.5)))
   (- (* im (* 0.5 (* re re))) im)
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -8.5e+16) || !(im <= 6.5)) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-8.5d+16)) .or. (.not. (im <= 6.5d0))) then
        tmp = (im * (0.5d0 * (re * re))) - im
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -8.5e+16) || !(im <= 6.5)) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -8.5e+16) or not (im <= 6.5):
		tmp = (im * (0.5 * (re * re))) - im
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -8.5e+16) || !(im <= 6.5))
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -8.5e+16) || ~((im <= 6.5)))
		tmp = (im * (0.5 * (re * re))) - im;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -8.5e+16], N[Not[LessEqual[im, 6.5]], $MachinePrecision]], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -8.5e16 or 6.5 < im

   1. Initial program 100.0%

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

      1. 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 5.8%

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

      1. associate-*r* 5.8%

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

      2. neg-mul-1 5.8%

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

   6. Simplified 5.8%

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

   7. Taylor expanded in re around 0 26.3%

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

      1. neg-mul-1 26.3%

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

      2. +-commutative 26.3%

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

      3. unsub-neg 26.3%

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

      4. associate-*r* 26.3%

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

      5. *-commutative 26.3%

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

      6. associate-*l* 26.3%

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

      7. *-commutative 26.3%

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

      8. unpow2 26.3%

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

   9. Simplified 26.3%

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

   if -8.5e16 < im < 6.5

   1. Initial program 14.5%

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

      1. neg-sub0 14.5%

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

   3. Simplified 14.5%

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

   4. Taylor expanded in im around 0 93.8%

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

      1. associate-*r* 93.8%

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

      2. neg-mul-1 93.8%

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

   6. Simplified 93.8%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.5 \cdot 10^{+16} \lor \neg \left(im \leq 6.5\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 11: 35.8% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.9 \cdot 10^{+158}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.9e+158) (- (* im (* 0.5 (* re re))) im) (* (* re re) 0.75)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.9e+158) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = (re * re) * 0.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3.9d+158) then
        tmp = (im * (0.5d0 * (re * re))) - im
    else
        tmp = (re * re) * 0.75d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.9e+158) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = (re * re) * 0.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.9e+158:
		tmp = (im * (0.5 * (re * re))) - im
	else:
		tmp = (re * re) * 0.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.9e+158)
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	else
		tmp = Float64(Float64(re * re) * 0.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.9e+158)
		tmp = (im * (0.5 * (re * re))) - im;
	else
		tmp = (re * re) * 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.9e+158], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.9e158

   1. Initial program 53.5%

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

      1. neg-sub0 53.5%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in im around 0 53.5%

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

      1. associate-*r* 53.5%

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

      2. neg-mul-1 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in re around 0 39.0%

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

      1. neg-mul-1 39.0%

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

      2. +-commutative 39.0%

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

      3. unsub-neg 39.0%

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

      4. associate-*r* 39.0%

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

      5. *-commutative 39.0%

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

      6. associate-*l* 39.0%

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

      7. *-commutative 39.0%

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

      8. unpow2 39.0%

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

   9. Simplified 39.0%

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

   if 3.9e158 < re

   1. Initial program 51.7%

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

      1. neg-sub0 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in re around 0 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 24.4%

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

      4. unpow2 24.4%

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

   6. Simplified 24.4%

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

   8. Applied egg-rr 34.2%

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

   9. Taylor expanded in re around inf 34.2%

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

       1. *-commutative 34.2%

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

       2. unpow2 34.2%

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

   11. Simplified 34.2%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.9 \cdot 10^{+158}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \]

Alternative 12: 32.0% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.35 \cdot 10^{+161}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.35e+161) (- im) (* (* re re) 0.75)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.35e+161) {
		tmp = -im;
	} else {
		tmp = (re * re) * 0.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.35d+161) then
        tmp = -im
    else
        tmp = (re * re) * 0.75d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.35e+161) {
		tmp = -im;
	} else {
		tmp = (re * re) * 0.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.35e+161:
		tmp = -im
	else:
		tmp = (re * re) * 0.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.35e+161)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(re * re) * 0.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.35e+161)
		tmp = -im;
	else
		tmp = (re * re) * 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.35e+161], (-im), N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.3499999999999999e161

   1. Initial program 53.4%

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

      1. neg-sub0 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in im around 0 53.7%

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

      1. associate-*r* 53.7%

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

      2. neg-mul-1 53.7%

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

   6. Simplified 53.7%

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

   7. Taylor expanded in re around 0 33.3%

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

      1. neg-mul-1 33.3%

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

   9. Simplified 33.3%

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

   if 1.3499999999999999e161 < re

   1. Initial program 51.9%

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

      1. neg-sub0 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in re around 0 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 25.9%

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

      4. unpow2 25.9%

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

   6. Simplified 25.9%

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

   8. Applied egg-rr 36.3%

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

   9. Taylor expanded in re around inf 36.3%

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

       1. *-commutative 36.3%

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

       2. unpow2 36.3%

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

   11. Simplified 36.3%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.35 \cdot 10^{+161}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \]

Alternative 13: 30.2% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. neg-sub0 53.2%

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

3. Simplified 53.2%

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

4. Taylor expanded in im around 0 53.9%

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

   1. associate-*r* 53.9%

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

   2. neg-mul-1 53.9%

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

6. Simplified 53.9%

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

7. Taylor expanded in re around 0 30.2%

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

   1. neg-mul-1 30.2%

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

9. Simplified 30.2%

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

10. Final simplification 30.2%

    \[\leadsto -im \]

Alternative 14: 2.9% accurate, 309.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -1.5

Julia

function code(re, im)
	return -1.5
end

MATLAB

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

Wolfram

code[re_, im_] := -1.5

Derivation

1. Initial program 53.2%

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

   1. neg-sub0 53.2%

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

3. Simplified 53.2%

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

4. Taylor expanded in re around 0 3.4%

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

   1. +-commutative 3.4%

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

   2. associate-*r* 3.4%

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

   3. distribute-rgt-out 37.8%

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

   4. unpow2 37.8%

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

6. Simplified 37.8%

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

8. Applied egg-rr 10.4%

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

9. Taylor expanded in re around 0 3.1%

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

10. Final simplification 3.1%

    \[\leadsto -1.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023279 
(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))))