math.sin on complex, imaginary part

Percentage Accurate: 54.0% → 99.8%

Time: 18.4s

Alternatives: 15

Speedup: 14.6×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.4 \lor \neg \left(t_0 \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - im \cdot \cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.4) (not (<= t_0 5e-5)))
     (* (* 0.5 (cos re)) t_0)
     (- (* -0.16666666666666666 (* (cos re) (pow im 3.0))) (* im (cos re))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.4) || !(t_0 <= 5e-5)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = (-0.16666666666666666 * (cos(re) * pow(im, 3.0))) - (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 <= (-0.4d0)) .or. (.not. (t_0 <= 5d-5))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = ((-0.16666666666666666d0) * (cos(re) * (im ** 3.0d0))) - (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 <= -0.4) || !(t_0 <= 5e-5)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = (-0.16666666666666666 * (Math.cos(re) * Math.pow(im, 3.0))) - (im * Math.cos(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.4) or not (t_0 <= 5e-5):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = (-0.16666666666666666 * (math.cos(re) * math.pow(im, 3.0))) - (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 <= -0.4) || !(t_0 <= 5e-5))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(Float64(-0.16666666666666666 * Float64(cos(re) * (im ^ 3.0))) - 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 <= -0.4) || ~((t_0 <= 5e-5)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = (-0.16666666666666666 * (cos(re) * (im ^ 3.0))) - (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, -0.4], N[Not[LessEqual[t$95$0, 5e-5]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(-0.16666666666666666 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 3.0], $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)) < -0.40000000000000002 or 5.00000000000000024e-5 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

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

      1. 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 -0.40000000000000002 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 5.00000000000000024e-5

   1. Initial program 7.3%

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

      1. neg-sub0 7.3%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.4 \lor \neg \left(e^{-im} - e^{im} \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - im \cdot \cos re\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.4 \lor \neg \left(t_0 \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.4) (not (<= t_0 5e-5)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.4) || !(t_0 <= 5e-5)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - 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 <= (-0.4d0)) .or. (.not. (t_0 <= 5d-5))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - 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 <= -0.4) || !(t_0 <= 5e-5)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.4) or not (t_0 <= 5e-5):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.4) || !(t_0 <= 5e-5))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.4) || ~((t_0 <= 5e-5)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - 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, -0.4], N[Not[LessEqual[t$95$0, 5e-5]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.40000000000000002 or 5.00000000000000024e-5 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

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

      1. 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 -0.40000000000000002 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 5.00000000000000024e-5

   1. Initial program 7.3%

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

      1. neg-sub0 7.3%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. distribute-rgt-out-- 99.8%

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

      6. *-commutative 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.4 \lor \neg \left(e^{-im} - e^{im} \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]

Alternative 3: 95.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+111} \lor \neg \left(im \leq -0.022 \lor \neg \left(im \leq 0.06\right) \land im \leq 6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.5e+111)
         (not (or (<= im -0.022) (and (not (<= im 0.06)) (<= im 6e+102)))))
   (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))
   (* (- (exp (- im)) (exp im)) (+ 0.5 (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.5e+111) || !((im <= -0.022) || (!(im <= 0.06) && (im <= 6e+102)))) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - 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 <= (-4.5d+111)) .or. (.not. (im <= (-0.022d0)) .or. (.not. (im <= 0.06d0)) .and. (im <= 6d+102))) then
        tmp = cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - 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 <= -4.5e+111) || !((im <= -0.022) || (!(im <= 0.06) && (im <= 6e+102)))) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - 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 <= -4.5e+111) or not ((im <= -0.022) or (not (im <= 0.06) and (im <= 6e+102))):
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - 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 <= -4.5e+111) || !((im <= -0.022) || (!(im <= 0.06) && (im <= 6e+102))))
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - 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 <= -4.5e+111) || ~(((im <= -0.022) || (~((im <= 0.06)) && (im <= 6e+102)))))
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - 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, -4.5e+111], N[Not[Or[LessEqual[im, -0.022], And[N[Not[LessEqual[im, 0.06]], $MachinePrecision], LessEqual[im, 6e+102]]]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $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 < -4.50000000000000001e111 or -0.021999999999999999 < im < 0.059999999999999998 or 5.9999999999999996e102 < im

   1. Initial program 41.8%

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

      1. neg-sub0 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in im around 0 99.9%

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

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

   if -4.50000000000000001e111 < im < -0.021999999999999999 or 0.059999999999999998 < im < 5.9999999999999996e102

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 6.1%

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

      1. +-commutative 6.1%

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

      2. associate-*r* 6.1%

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

      3. distribute-rgt-out 83.7%

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

      4. unpow2 83.7%

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

   6. Simplified 83.7%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+111} \lor \neg \left(im \leq -0.022 \lor \neg \left(im \leq 0.06\right) \land im \leq 6 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 4: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{+102} \lor \neg \left(im \leq -0.0155\right) \land \left(im \leq 0.024 \lor \neg \left(im \leq 2.7 \cdot 10^{+99}\right)\right):\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -5.5e+102)
         (and (not (<= im -0.0155)) (or (<= im 0.024) (not (<= im 2.7e+99)))))
   (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))
   (* 0.5 (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -5.5e+102) || (!(im <= -0.0155) && ((im <= 0.024) || !(im <= 2.7e+99)))) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - 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 <= (-0.0155d0))) .and. (im <= 0.024d0) .or. (.not. (im <= 2.7d+99))) then
        tmp = cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - 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 <= -0.0155) && ((im <= 0.024) || !(im <= 2.7e+99)))) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - 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 <= -0.0155) and ((im <= 0.024) or not (im <= 2.7e+99))):
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - 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 <= -0.0155) && ((im <= 0.024) || !(im <= 2.7e+99))))
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - 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 <= -0.0155)) && ((im <= 0.024) || ~((im <= 2.7e+99)))))
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - 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], And[N[Not[LessEqual[im, -0.0155]], $MachinePrecision], Or[LessEqual[im, 0.024], N[Not[LessEqual[im, 2.7e+99]], $MachinePrecision]]]], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $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 -0.0155 < im < 0.024 or 2.69999999999999989e99 < im

   1. Initial program 43.2%

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

      1. neg-sub0 43.2%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in im around 0 99.5%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   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 < -0.0155 or 0.024 < im < 2.69999999999999989e99

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 72.7%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{+102} \lor \neg \left(im \leq -0.0155\right) \land \left(im \leq 0.024 \lor \neg \left(im \leq 2.7 \cdot 10^{+99}\right)\right):\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 5: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \left(\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right) - im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{if}\;im \leq -1.5 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.00011:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0012:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3} - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1
         (*
          (- (* (* im im) (* im -0.16666666666666666)) im)
          (+ 1.0 (* re (* re -0.5))))))
   (if (<= im -1.5e+103)
     t_1
     (if (<= im -0.00011)
       t_0
       (if (<= im 0.0012)
         (* (cos re) (- im))
         (if (<= im 6.5e+51)
           t_0
           (if (<= im 1.8e+133)
             t_1
             (- (* -0.16666666666666666 (pow im 3.0)) im))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = (((im * im) * (im * -0.16666666666666666)) - im) * (1.0 + (re * (re * -0.5)));
	double tmp;
	if (im <= -1.5e+103) {
		tmp = t_1;
	} else if (im <= -0.00011) {
		tmp = t_0;
	} else if (im <= 0.0012) {
		tmp = cos(re) * -im;
	} else if (im <= 6.5e+51) {
		tmp = t_0;
	} else if (im <= 1.8e+133) {
		tmp = t_1;
	} else {
		tmp = (-0.16666666666666666 * pow(im, 3.0)) - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (((im * im) * (im * (-0.16666666666666666d0))) - im) * (1.0d0 + (re * (re * (-0.5d0))))
    if (im <= (-1.5d+103)) then
        tmp = t_1
    else if (im <= (-0.00011d0)) then
        tmp = t_0
    else if (im <= 0.0012d0) then
        tmp = cos(re) * -im
    else if (im <= 6.5d+51) then
        tmp = t_0
    else if (im <= 1.8d+133) then
        tmp = t_1
    else
        tmp = ((-0.16666666666666666d0) * (im ** 3.0d0)) - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = (((im * im) * (im * -0.16666666666666666)) - im) * (1.0 + (re * (re * -0.5)));
	double tmp;
	if (im <= -1.5e+103) {
		tmp = t_1;
	} else if (im <= -0.00011) {
		tmp = t_0;
	} else if (im <= 0.0012) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 6.5e+51) {
		tmp = t_0;
	} else if (im <= 1.8e+133) {
		tmp = t_1;
	} else {
		tmp = (-0.16666666666666666 * Math.pow(im, 3.0)) - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = (((im * im) * (im * -0.16666666666666666)) - im) * (1.0 + (re * (re * -0.5)))
	tmp = 0
	if im <= -1.5e+103:
		tmp = t_1
	elif im <= -0.00011:
		tmp = t_0
	elif im <= 0.0012:
		tmp = math.cos(re) * -im
	elif im <= 6.5e+51:
		tmp = t_0
	elif im <= 1.8e+133:
		tmp = t_1
	else:
		tmp = (-0.16666666666666666 * math.pow(im, 3.0)) - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(Float64(Float64(Float64(im * im) * Float64(im * -0.16666666666666666)) - im) * Float64(1.0 + Float64(re * Float64(re * -0.5))))
	tmp = 0.0
	if (im <= -1.5e+103)
		tmp = t_1;
	elseif (im <= -0.00011)
		tmp = t_0;
	elseif (im <= 0.0012)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 6.5e+51)
		tmp = t_0;
	elseif (im <= 1.8e+133)
		tmp = t_1;
	else
		tmp = Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = (((im * im) * (im * -0.16666666666666666)) - im) * (1.0 + (re * (re * -0.5)));
	tmp = 0.0;
	if (im <= -1.5e+103)
		tmp = t_1;
	elseif (im <= -0.00011)
		tmp = t_0;
	elseif (im <= 0.0012)
		tmp = cos(re) * -im;
	elseif (im <= 6.5e+51)
		tmp = t_0;
	elseif (im <= 1.8e+133)
		tmp = t_1;
	else
		tmp = (-0.16666666666666666 * (im ^ 3.0)) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision] * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.5e+103], t$95$1, If[LessEqual[im, -0.00011], t$95$0, If[LessEqual[im, 0.0012], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 6.5e+51], t$95$0, If[LessEqual[im, 1.8e+133], t$95$1, N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.5e103 or 6.5e51 < im < 1.79999999999999989e133

   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 90.2%

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

      1. +-commutative 90.2%

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

      2. mul-1-neg 90.2%

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

      3. unsub-neg 90.2%

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

      4. associate-*r* 90.2%

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

      5. distribute-rgt-out-- 90.2%

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

      6. *-commutative 90.2%

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

   6. Simplified 90.2%

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

   7. Taylor expanded in re around 0 7.3%

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

      1. associate--l+ 7.3%

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

      2. associate-*r* 7.3%

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

      3. distribute-lft1-in 82.7%

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

      4. *-commutative 82.7%

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

      5. *-inverses 82.7%

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

      6. +-commutative 82.7%

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

      7. *-inverses 82.7%

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

      8. unpow2 82.7%

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

      9. associate-*r* 82.7%

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

      10. *-commutative 82.7%

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

      11. *-commutative 82.7%

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

   9. Simplified 82.7%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)} \]
   10. Step-by-step derivation

       1. sub-neg 82.7%

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

       2. *-commutative 82.7%

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

       3. unpow3 82.7%

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

       4. associate-*l* 82.7%

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

       5. fma-def 82.7%

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

   11. Applied egg-rr 82.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, -im\right)} \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right) \]
   12. Step-by-step derivation

       1. fma-udef 82.7%

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

   13. Applied egg-rr 82.7%

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

   if -1.5e103 < im < -1.10000000000000004e-4 or 0.00119999999999999989 < im < 6.5e51

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 76.9%

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

   if -1.10000000000000004e-4 < im < 0.00119999999999999989

   1. Initial program 7.3%

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

      1. neg-sub0 7.3%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in im around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. neg-mul-1 99.6%

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

   6. Simplified 99.6%

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

   if 1.79999999999999989e133 < 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 0 100.0%

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

   5. Taylor expanded in re around 0 90.0%

      \[\leadsto \color{blue}{-1 \cdot im + -0.16666666666666666 \cdot {im}^{3}} \]
   6. Step-by-step derivation

      1. +-commutative 90.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} + -1 \cdot im} \]

      2. neg-mul-1 90.0%

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

      3. sub-neg 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+103}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right) - im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{elif}\;im \leq -0.00011:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.0012:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+133}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right) - im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3} - im\\ \end{array} \]

Alternative 6: 77.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6.2 \cdot 10^{+15} \lor \neg \left(im \leq 4.8 \cdot 10^{-20}\right):\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right) - im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -6.2e+15) (not (<= im 4.8e-20)))
   (*
    (- (* (* im im) (* im -0.16666666666666666)) im)
    (+ 1.0 (* re (* re -0.5))))
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -6.2e+15) || !(im <= 4.8e-20)) {
		tmp = (((im * im) * (im * -0.16666666666666666)) - im) * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = cos(re) * -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-6.2d+15)) .or. (.not. (im <= 4.8d-20))) then
        tmp = (((im * im) * (im * (-0.16666666666666666d0))) - im) * (1.0d0 + (re * (re * (-0.5d0))))
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -6.2e+15) || !(im <= 4.8e-20)) {
		tmp = (((im * im) * (im * -0.16666666666666666)) - im) * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -6.2e+15) or not (im <= 4.8e-20):
		tmp = (((im * im) * (im * -0.16666666666666666)) - im) * (1.0 + (re * (re * -0.5)))
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -6.2e+15) || !(im <= 4.8e-20))
		tmp = Float64(Float64(Float64(Float64(im * im) * Float64(im * -0.16666666666666666)) - im) * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -6.2e+15) || ~((im <= 4.8e-20)))
		tmp = (((im * im) * (im * -0.16666666666666666)) - im) * (1.0 + (re * (re * -0.5)));
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -6.2e+15], N[Not[LessEqual[im, 4.8e-20]], $MachinePrecision]], N[(N[(N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision] * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -6.2e15 or 4.79999999999999986e-20 < im

   1. Initial program 99.2%

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

      1. neg-sub0 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in im around 0 67.9%

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

      1. +-commutative 67.9%

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

      2. mul-1-neg 67.9%

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

      3. unsub-neg 67.9%

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

      4. associate-*r* 67.9%

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

      5. distribute-rgt-out-- 67.9%

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

      6. *-commutative 67.9%

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

   6. Simplified 67.9%

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

   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 63.4%

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

      4. *-commutative 63.4%

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

      5. *-inverses 63.4%

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

      6. +-commutative 63.4%

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

      7. *-inverses 63.4%

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

      8. unpow2 63.4%

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

      9. associate-*r* 63.4%

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

      10. *-commutative 63.4%

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

      11. *-commutative 63.4%

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

   9. Simplified 63.4%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)} \]
   10. Step-by-step derivation

       1. sub-neg 63.4%

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

       2. *-commutative 63.4%

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

       3. unpow3 63.4%

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

       4. associate-*l* 63.4%

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

       5. fma-def 63.4%

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

   11. Applied egg-rr 63.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, -im\right)} \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right) \]
   12. Step-by-step derivation

       1. fma-udef 63.4%

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

   13. Applied egg-rr 63.4%

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

   if -6.2e15 < im < 4.79999999999999986e-20

   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 98.1%

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

      1. associate-*r* 98.1%

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

      2. neg-mul-1 98.1%

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

   6. Simplified 98.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.2 \cdot 10^{+15} \lor \neg \left(im \leq 4.8 \cdot 10^{-20}\right):\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right) - im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 7: 44.4% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(0.5 \cdot re\right)\\ t_1 := \frac{im \cdot im}{im \cdot \left(-1 - t_0\right)}\\ \mathbf{if}\;im \leq -1.3 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -7.2 \cdot 10^{+19}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 1.4:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+155}:\\ \;\;\;\;\left(im + 1.1666666666666667\right) \cdot \left(-1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* 0.5 re))) (t_1 (/ (* im im) (* im (- -1.0 t_0)))))
   (if (<= im -1.3e+170)
     t_1
     (if (<= im -7.2e+19)
       (- (* im (* 0.5 (* re re))) im)
       (if (<= im 1.4)
         (- im)
         (if (<= im 1.6e+155)
           (* (+ im 1.1666666666666667) (+ -1.0 t_0))
           t_1))))))

C

double code(double re, double im) {
	double t_0 = re * (0.5 * re);
	double t_1 = (im * im) / (im * (-1.0 - t_0));
	double tmp;
	if (im <= -1.3e+170) {
		tmp = t_1;
	} else if (im <= -7.2e+19) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else if (im <= 1.4) {
		tmp = -im;
	} else if (im <= 1.6e+155) {
		tmp = (im + 1.1666666666666667) * (-1.0 + t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * (0.5d0 * re)
    t_1 = (im * im) / (im * ((-1.0d0) - t_0))
    if (im <= (-1.3d+170)) then
        tmp = t_1
    else if (im <= (-7.2d+19)) then
        tmp = (im * (0.5d0 * (re * re))) - im
    else if (im <= 1.4d0) then
        tmp = -im
    else if (im <= 1.6d+155) then
        tmp = (im + 1.1666666666666667d0) * ((-1.0d0) + t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (0.5 * re);
	double t_1 = (im * im) / (im * (-1.0 - t_0));
	double tmp;
	if (im <= -1.3e+170) {
		tmp = t_1;
	} else if (im <= -7.2e+19) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else if (im <= 1.4) {
		tmp = -im;
	} else if (im <= 1.6e+155) {
		tmp = (im + 1.1666666666666667) * (-1.0 + t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (0.5 * re)
	t_1 = (im * im) / (im * (-1.0 - t_0))
	tmp = 0
	if im <= -1.3e+170:
		tmp = t_1
	elif im <= -7.2e+19:
		tmp = (im * (0.5 * (re * re))) - im
	elif im <= 1.4:
		tmp = -im
	elif im <= 1.6e+155:
		tmp = (im + 1.1666666666666667) * (-1.0 + t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(0.5 * re))
	t_1 = Float64(Float64(im * im) / Float64(im * Float64(-1.0 - t_0)))
	tmp = 0.0
	if (im <= -1.3e+170)
		tmp = t_1;
	elseif (im <= -7.2e+19)
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	elseif (im <= 1.4)
		tmp = Float64(-im);
	elseif (im <= 1.6e+155)
		tmp = Float64(Float64(im + 1.1666666666666667) * Float64(-1.0 + t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (0.5 * re);
	t_1 = (im * im) / (im * (-1.0 - t_0));
	tmp = 0.0;
	if (im <= -1.3e+170)
		tmp = t_1;
	elseif (im <= -7.2e+19)
		tmp = (im * (0.5 * (re * re))) - im;
	elseif (im <= 1.4)
		tmp = -im;
	elseif (im <= 1.6e+155)
		tmp = (im + 1.1666666666666667) * (-1.0 + t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im * im), $MachinePrecision] / N[(im * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.3e+170], t$95$1, If[LessEqual[im, -7.2e+19], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, 1.4], (-im), If[LessEqual[im, 1.6e+155], N[(N[(im + 1.1666666666666667), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.2999999999999999e170 or 1.60000000000000006e155 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 85.2%

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

      4. unpow2 85.2%

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

   6. Simplified 85.2%

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

   7. Taylor expanded in im around 0 14.4%

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

      1. distribute-lft-in 14.4%

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

      2. flip-+ 57.4%

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

   9. Applied egg-rr 57.4%

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

   11. Simplified 75.9%

       \[\leadsto \color{blue}{\frac{im \cdot \left(im - re \cdot \left(\left(re \cdot 0.5\right) \cdot \left(re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right)\right)\right)\right)}{im \cdot \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)}} \]

   12. Taylor expanded in re around 0 70.4%

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

       1. unpow2 70.4%

          \[\leadsto \frac{\color{blue}{im \cdot im}}{im \cdot \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)} \]

   14. Simplified 70.4%

       \[\leadsto \frac{\color{blue}{im \cdot im}}{im \cdot \left(-1 - re \cdot \left(re \cdot 0.5\right)\right)} \]

   if -1.2999999999999999e170 < im < -7.2e19

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.9%

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

      1. associate-*r* 3.9%

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

      2. neg-mul-1 3.9%

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

   6. Simplified 3.9%

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

   7. Taylor expanded in re around 0 26.8%

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

      1. +-commutative 26.8%

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

      2. neg-mul-1 26.8%

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

      3. unsub-neg 26.8%

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

      4. *-commutative 26.8%

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

      5. associate-*l* 26.8%

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

      6. *-commutative 26.8%

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

      7. unpow2 26.8%

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

   9. Simplified 26.8%

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

   if -7.2e19 < im < 1.3999999999999999

   1. Initial program 10.7%

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

      1. neg-sub0 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in im around 0 96.2%

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

      1. associate-*r* 96.2%

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

      2. neg-mul-1 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in re around 0 55.0%

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

      1. neg-mul-1 55.0%

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

   9. Simplified 55.0%

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

   if 1.3999999999999999 < im < 1.60000000000000006e155

   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 47.8%

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

      1. +-commutative 47.8%

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

      2. mul-1-neg 47.8%

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

      3. unsub-neg 47.8%

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

      4. associate-*r* 47.8%

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

      5. distribute-rgt-out-- 47.8%

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

      6. *-commutative 47.8%

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

   6. Simplified 47.8%

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

   8. Applied egg-rr 4.9%

      \[\leadsto \cos re \cdot \left(\color{blue}{-1.1666666666666667} - im\right) \]

   9. Taylor expanded in re around 0 24.2%

      \[\leadsto \color{blue}{-1 \cdot \left(1.1666666666666667 + im\right) + 0.5 \cdot \left({re}^{2} \cdot \left(1.1666666666666667 + im\right)\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in 24.2%

          \[\leadsto \color{blue}{\left(-1 \cdot 1.1666666666666667 + -1 \cdot im\right)} + 0.5 \cdot \left({re}^{2} \cdot \left(1.1666666666666667 + im\right)\right) \]

       2. metadata-eval 24.2%

          \[\leadsto \left(\color{blue}{-1.1666666666666667} + -1 \cdot im\right) + 0.5 \cdot \left({re}^{2} \cdot \left(1.1666666666666667 + im\right)\right) \]

       3. neg-mul-1 24.2%

          \[\leadsto \left(-1.1666666666666667 + \color{blue}{\left(-im\right)}\right) + 0.5 \cdot \left({re}^{2} \cdot \left(1.1666666666666667 + im\right)\right) \]

       4. sub-neg 24.2%

          \[\leadsto \color{blue}{\left(-1.1666666666666667 - im\right)} + 0.5 \cdot \left({re}^{2} \cdot \left(1.1666666666666667 + im\right)\right) \]

       5. +-commutative 24.2%

          \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot \left(1.1666666666666667 + im\right)\right) + \left(-1.1666666666666667 - im\right)} \]

       6. unpow2 24.2%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(1.1666666666666667 + im\right)\right) + \left(-1.1666666666666667 - im\right) \]

       7. associate-*r* 24.2%

          \[\leadsto \color{blue}{\left(0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(1.1666666666666667 + im\right)} + \left(-1.1666666666666667 - im\right) \]

       8. *-commutative 24.2%

          \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot 0.5\right)} \cdot \left(1.1666666666666667 + im\right) + \left(-1.1666666666666667 - im\right) \]

       9. associate-*r* 24.2%

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \cdot \left(1.1666666666666667 + im\right) + \left(-1.1666666666666667 - im\right) \]

       10. sub-neg 24.2%

           \[\leadsto \left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(1.1666666666666667 + im\right) + \color{blue}{\left(-1.1666666666666667 + \left(-im\right)\right)} \]

       11. metadata-eval 24.2%

           \[\leadsto \left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(1.1666666666666667 + im\right) + \left(\color{blue}{-1 \cdot 1.1666666666666667} + \left(-im\right)\right) \]

       12. neg-mul-1 24.2%

           \[\leadsto \left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(1.1666666666666667 + im\right) + \left(-1 \cdot 1.1666666666666667 + \color{blue}{-1 \cdot im}\right) \]

       13. distribute-lft-in 24.2%

           \[\leadsto \left(re \cdot \left(re \cdot 0.5\right)\right) \cdot \left(1.1666666666666667 + im\right) + \color{blue}{-1 \cdot \left(1.1666666666666667 + im\right)} \]

       14. distribute-rgt-out 24.2%

           \[\leadsto \color{blue}{\left(1.1666666666666667 + im\right) \cdot \left(re \cdot \left(re \cdot 0.5\right) + -1\right)} \]

       15. +-commutative 24.2%

           \[\leadsto \color{blue}{\left(im + 1.1666666666666667\right)} \cdot \left(re \cdot \left(re \cdot 0.5\right) + -1\right) \]

   11. Simplified 24.2%

       \[\leadsto \color{blue}{\left(im + 1.1666666666666667\right) \cdot \left(re \cdot \left(re \cdot 0.5\right) + -1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+170}:\\ \;\;\;\;\frac{im \cdot im}{im \cdot \left(-1 - re \cdot \left(0.5 \cdot re\right)\right)}\\ \mathbf{elif}\;im \leq -7.2 \cdot 10^{+19}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 1.4:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+155}:\\ \;\;\;\;\left(im + 1.1666666666666667\right) \cdot \left(-1 + re \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot im}{im \cdot \left(-1 - re \cdot \left(0.5 \cdot re\right)\right)}\\ \end{array} \]

Alternative 8: 55.6% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+15} \lor \neg \left(im \leq 1.7 \cdot 10^{-24}\right):\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right) - im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.1e+15) (not (<= im 1.7e-24)))
   (*
    (- (* (* im im) (* im -0.16666666666666666)) im)
    (+ 1.0 (* re (* re -0.5))))
   (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.1e+15) || !(im <= 1.7e-24)) {
		tmp = (((im * im) * (im * -0.16666666666666666)) - im) * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-2.1d+15)) .or. (.not. (im <= 1.7d-24))) then
        tmp = (((im * im) * (im * (-0.16666666666666666d0))) - im) * (1.0d0 + (re * (re * (-0.5d0))))
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.1e+15) || !(im <= 1.7e-24)) {
		tmp = (((im * im) * (im * -0.16666666666666666)) - im) * (1.0 + (re * (re * -0.5)));
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.1e+15) or not (im <= 1.7e-24):
		tmp = (((im * im) * (im * -0.16666666666666666)) - im) * (1.0 + (re * (re * -0.5)))
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.1e+15) || !(im <= 1.7e-24))
		tmp = Float64(Float64(Float64(Float64(im * im) * Float64(im * -0.16666666666666666)) - im) * Float64(1.0 + Float64(re * Float64(re * -0.5))));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.1e+15) || ~((im <= 1.7e-24)))
		tmp = (((im * im) * (im * -0.16666666666666666)) - im) * (1.0 + (re * (re * -0.5)));
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.1e+15], N[Not[LessEqual[im, 1.7e-24]], $MachinePrecision]], N[(N[(N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision] * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -2.1e15 or 1.69999999999999996e-24 < im

   1. Initial program 98.4%

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

      1. neg-sub0 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in im around 0 68.2%

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

      1. +-commutative 68.2%

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

      2. mul-1-neg 68.2%

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

      3. unsub-neg 68.2%

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

      4. associate-*r* 68.2%

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

      5. distribute-rgt-out-- 68.2%

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

      6. *-commutative 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in re around 0 8.9%

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

      1. associate--l+ 8.9%

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

      2. associate-*r* 8.9%

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

      3. distribute-lft1-in 62.9%

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

      4. *-commutative 62.9%

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

      5. *-inverses 62.9%

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

      6. +-commutative 62.9%

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

      7. *-inverses 62.9%

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

      8. unpow2 62.9%

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

      9. associate-*r* 62.9%

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

      10. *-commutative 62.9%

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

      11. *-commutative 62.9%

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

   9. Simplified 62.9%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)} \]
   10. Step-by-step derivation

       1. sub-neg 62.9%

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

       2. *-commutative 62.9%

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

       3. unpow3 62.9%

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

       4. associate-*l* 62.9%

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

       5. fma-def 62.9%

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

   11. Applied egg-rr 62.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, -im\right)} \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right) \]
   12. Step-by-step derivation

       1. fma-udef 62.9%

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

   13. Applied egg-rr 62.9%

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

   if -2.1e15 < im < 1.69999999999999996e-24

   1. Initial program 8.8%

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

      1. neg-sub0 8.8%

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

   3. Simplified 8.8%

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

   4. Taylor expanded in im around 0 98.1%

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

      1. associate-*r* 98.1%

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

      2. neg-mul-1 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in re around 0 56.1%

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

      1. neg-mul-1 56.1%

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

   9. Simplified 56.1%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+15} \lor \neg \left(im \leq 1.7 \cdot 10^{-24}\right):\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot -0.16666666666666666\right) - im\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 9: 32.1% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.7 \cdot 10^{+110}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.7e+110) (- im) (* 0.5 (* im (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.7e+110) {
		tmp = -im;
	} else {
		tmp = 0.5 * (im * (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 (re <= 5.7d+110) then
        tmp = -im
    else
        tmp = 0.5d0 * (im * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.7e+110) {
		tmp = -im;
	} else {
		tmp = 0.5 * (im * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.7e+110:
		tmp = -im
	else:
		tmp = 0.5 * (im * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.7e+110)
		tmp = Float64(-im);
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.7e+110)
		tmp = -im;
	else
		tmp = 0.5 * (im * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.7e+110], (-im), N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.7000000000000002e110

   1. Initial program 55.1%

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

      1. neg-sub0 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in im around 0 51.2%

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

      1. associate-*r* 51.2%

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

      2. neg-mul-1 51.2%

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

   6. Simplified 51.2%

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

   7. Taylor expanded in re around 0 35.3%

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

      1. neg-mul-1 35.3%

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

   9. Simplified 35.3%

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

   if 5.7000000000000002e110 < re

   1. Initial program 41.5%

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

      1. neg-sub0 41.5%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in re around 0 1.1%

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

      1. +-commutative 1.1%

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

      2. associate-*r* 1.1%

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

      3. distribute-rgt-out 23.0%

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

      4. unpow2 23.0%

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

   6. Simplified 23.0%

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

   7. Taylor expanded in im around 0 23.2%

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

   8. Taylor expanded in re around inf 23.2%

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

      1. unpow2 23.2%

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

   10. Simplified 23.2%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.7 \cdot 10^{+110}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 10: 32.2% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.7 \cdot 10^{+110}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.7e+110) (- im) (* re (* re (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.7e+110) {
		tmp = -im;
	} else {
		tmp = re * (re * (im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5.7d+110) then
        tmp = -im
    else
        tmp = re * (re * (im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.7e+110) {
		tmp = -im;
	} else {
		tmp = re * (re * (im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.7e+110:
		tmp = -im
	else:
		tmp = re * (re * (im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.7e+110)
		tmp = Float64(-im);
	else
		tmp = Float64(re * Float64(re * Float64(im * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.7e+110)
		tmp = -im;
	else
		tmp = re * (re * (im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.7e+110], (-im), N[(re * N[(re * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.7000000000000002e110

   1. Initial program 55.1%

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

      1. neg-sub0 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in im around 0 51.2%

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

      1. associate-*r* 51.2%

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

      2. neg-mul-1 51.2%

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

   6. Simplified 51.2%

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

   7. Taylor expanded in re around 0 35.3%

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

      1. neg-mul-1 35.3%

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

   9. Simplified 35.3%

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

   if 5.7000000000000002e110 < re

   1. Initial program 41.5%

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

      1. neg-sub0 41.5%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in re around 0 1.1%

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

      1. +-commutative 1.1%

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

      2. associate-*r* 1.1%

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

      3. distribute-rgt-out 23.0%

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

      4. unpow2 23.0%

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

   6. Simplified 23.0%

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

   7. Taylor expanded in im around 0 23.2%

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

   8. Taylor expanded in re around inf 23.2%

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

      1. *-commutative 23.2%

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

      2. unpow2 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-*r* 23.2%

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

      5. associate-*l* 23.3%

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

   10. Simplified 23.3%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.7 \cdot 10^{+110}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 11: 35.9% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. neg-sub0 52.9%

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

3. Simplified 52.9%

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

4. Taylor expanded in im around 0 53.3%

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

   1. associate-*r* 53.3%

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

   2. neg-mul-1 53.3%

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

6. Simplified 53.3%

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

7. Taylor expanded in re around 0 36.3%

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

   1. +-commutative 36.3%

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

   2. neg-mul-1 36.3%

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

   3. unsub-neg 36.3%

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

   4. *-commutative 36.3%

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

   5. associate-*l* 36.3%

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

   6. *-commutative 36.3%

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

   7. unpow2 36.3%

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

9. Simplified 36.3%

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

10. Final simplification 36.3%

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

Alternative 12: 35.9% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. neg-sub0 52.9%

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

3. Simplified 52.9%

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

4. Taylor expanded in re around 0 3.8%

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

   1. +-commutative 3.8%

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

   2. associate-*r* 3.8%

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

   3. distribute-rgt-out 43.6%

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

   4. unpow2 43.6%

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

6. Simplified 43.6%

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

7. Taylor expanded in im around 0 36.3%

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

8. Taylor expanded in im around 0 36.3%

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

   1. distribute-lft-in 36.3%

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

   2. unpow2 36.3%

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

   3. *-commutative 36.3%

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

   4. associate-*r* 36.3%

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

   5. distribute-lft-in 36.3%

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

   6. *-commutative 36.3%

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

   7. associate-*r* 36.3%

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

   8. metadata-eval 36.3%

      \[\leadsto \color{blue}{-1} \cdot im + -2 \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right) \]

   9. neg-mul-1 36.3%

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

   10. associate-*r* 36.3%

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

   11. *-commutative 36.3%

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

   12. associate-*r* 36.3%

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

   13. *-commutative 36.3%

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

   14. associate-*r* 36.3%

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

   15. associate-*l* 36.3%

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

   16. metadata-eval 36.3%

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

   17. *-commutative 36.3%

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

   18. associate-*r* 36.3%

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

   19. unpow2 36.3%

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

   20. +-commutative 36.3%

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

   21. unsub-neg 36.3%

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

10. Simplified 36.4%

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

11. Final simplification 36.4%

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

Alternative 13: 31.6% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{+171}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 4.4e+171) (- im) (* re re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.4e+171) {
		tmp = -im;
	} else {
		tmp = 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 (re <= 4.4d+171) then
        tmp = -im
    else
        tmp = re * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.4e+171) {
		tmp = -im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.4e+171:
		tmp = -im
	else:
		tmp = re * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.4e+171)
		tmp = Float64(-im);
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.4e+171)
		tmp = -im;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.4e+171], (-im), N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.3999999999999999e171

   1. Initial program 53.0%

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

      1. neg-sub0 53.0%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in im around 0 53.1%

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

      1. associate-*r* 53.1%

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

      2. neg-mul-1 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in re around 0 33.8%

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

      1. neg-mul-1 33.8%

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

   9. Simplified 33.8%

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

   if 4.3999999999999999e171 < 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 25.1%

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

      4. unpow2 25.1%

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

   6. Simplified 25.1%

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

   8. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, -re\right)} \]
   9. Step-by-step derivation

      1. fma-neg 25.8%

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

   10. Simplified 25.8%

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

   11. Taylor expanded in re around inf 25.8%

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

       1. unpow2 25.8%

          \[\leadsto \color{blue}{re \cdot re} \]

   13. Simplified 25.8%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{+171}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 14: 29.5% 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 52.9%

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

   1. neg-sub0 52.9%

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

3. Simplified 52.9%

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

4. Taylor expanded in im around 0 53.3%

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

   1. associate-*r* 53.3%

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

   2. neg-mul-1 53.3%

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

6. Simplified 53.3%

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

7. Taylor expanded in re around 0 31.0%

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

   1. neg-mul-1 31.0%

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

9. Simplified 31.0%

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

10. Final simplification 31.0%

    \[\leadsto -im \]

Alternative 15: 3.0% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 52.9%

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

   1. neg-sub0 52.9%

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

3. Simplified 52.9%

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

4. Taylor expanded in re around 0 3.8%

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

   1. +-commutative 3.8%

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

   2. associate-*r* 3.8%

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

   3. distribute-rgt-out 43.6%

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

   4. unpow2 43.6%

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

6. Simplified 43.6%

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

8. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{\left|re\right|} \]
9. Step-by-step derivation

   1. unpow1 2.9%

      \[\leadsto \left|\color{blue}{{re}^{1}}\right| \]

   2. *-inverses 2.9%

      \[\leadsto \left|{re}^{\color{blue}{\left(\frac{e^{\cos re}}{e^{\cos re}}\right)}}\right| \]

   3. sqr-pow 1.7%

      \[\leadsto \left|\color{blue}{{re}^{\left(\frac{\frac{e^{\cos re}}{e^{\cos re}}}{2}\right)} \cdot {re}^{\left(\frac{\frac{e^{\cos re}}{e^{\cos re}}}{2}\right)}}\right| \]

   4. fabs-sqr 1.7%

      \[\leadsto \color{blue}{{re}^{\left(\frac{\frac{e^{\cos re}}{e^{\cos re}}}{2}\right)} \cdot {re}^{\left(\frac{\frac{e^{\cos re}}{e^{\cos re}}}{2}\right)}} \]

   5. sqr-pow 3.1%

      \[\leadsto \color{blue}{{re}^{\left(\frac{e^{\cos re}}{e^{\cos re}}\right)}} \]

   6. *-inverses 3.1%

      \[\leadsto {re}^{\color{blue}{1}} \]

   7. unpow1 3.1%

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

10. Simplified 3.1%

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

11. Final simplification 3.1%

    \[\leadsto re \]

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 2023275 
(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))))