math.sin on complex, imaginary part

Percentage Accurate: 54.3% → 99.5%

Time: 10.4s

Alternatives: 16

Speedup: 2.9×

• 49.5% 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: 54.3% 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: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], 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[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 5.00000000000000024e-5

   1. Initial program 8.9%

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

      1. sub0-neg 8.9%

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

   3. Simplified 8.9%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

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

Alternative 2: 94.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -8.6 \cdot 10^{+89} \lor \neg \left(im \leq -2.3 \cdot 10^{+15} \lor \neg \left(im \leq 0.055\right) \land im \leq 5.7 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -8.6e+89)
         (not
          (or (<= im -2.3e+15) (and (not (<= im 0.055)) (<= im 5.7e+102)))))
   (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
   (* 0.5 (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -8.6e+89) || !((im <= -2.3e+15) || (!(im <= 0.055) && (im <= 5.7e+102)))) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-8.6d+89)) .or. (.not. (im <= (-2.3d+15)) .or. (.not. (im <= 0.055d0)) .and. (im <= 5.7d+102))) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = 0.5d0 * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -8.6e+89) || !((im <= -2.3e+15) || (!(im <= 0.055) && (im <= 5.7e+102)))) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -8.6e+89) or not ((im <= -2.3e+15) or (not (im <= 0.055) and (im <= 5.7e+102))):
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -8.6e+89) || !((im <= -2.3e+15) || (!(im <= 0.055) && (im <= 5.7e+102))))
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -8.6e+89) || ~(((im <= -2.3e+15) || (~((im <= 0.055)) && (im <= 5.7e+102)))))
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = 0.5 * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -8.6e+89], N[Not[Or[LessEqual[im, -2.3e+15], And[N[Not[LessEqual[im, 0.055]], $MachinePrecision], LessEqual[im, 5.7e+102]]]], $MachinePrecision]], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -8.6000000000000003e89 or -2.3e15 < im < 0.0550000000000000003 or 5.6999999999999999e102 < im

   1. Initial program 47.6%

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

      1. sub0-neg 47.6%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in im around 0 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

      3. *-commutative 98.5%

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

      4. associate-*l* 98.5%

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

      5. distribute-lft-out-- 98.5%

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

   6. Simplified 98.5%

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

   if -8.6000000000000003e89 < im < -2.3e15 or 0.0550000000000000003 < im < 5.6999999999999999e102

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 83.9%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.6 \cdot 10^{+89} \lor \neg \left(im \leq -2.3 \cdot 10^{+15} \lor \neg \left(im \leq 0.055\right) \land im \leq 5.7 \cdot 10^{+102}\right):\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 3: 93.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00027:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (/ (* (cos re) (- 9.0 (* im im))) (+ im -3.0))))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -1.7e-5)
       t_0
       (if (<= im 0.00027)
         (* (cos re) (- im))
         (if (<= im 1.32e+154) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -1.7e-5) {
		tmp = t_0;
	} else if (im <= 0.00027) {
		tmp = cos(re) * -im;
	} else if (im <= 1.32e+154) {
		tmp = 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 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (cos(re) * (9.0d0 - (im * im))) / (im + (-3.0d0))
    if (im <= (-1.35d+154)) then
        tmp = t_1
    else if (im <= (-1.7d-5)) then
        tmp = t_0
    else if (im <= 0.00027d0) then
        tmp = cos(re) * -im
    else if (im <= 1.32d+154) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = (Math.cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -1.7e-5) {
		tmp = t_0;
	} else if (im <= 0.00027) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.32e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = (math.cos(re) * (9.0 - (im * im))) / (im + -3.0)
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_1
	elif im <= -1.7e-5:
		tmp = t_0
	elif im <= 0.00027:
		tmp = math.cos(re) * -im
	elif im <= 1.32e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(Float64(cos(re) * Float64(9.0 - Float64(im * im))) / Float64(im + -3.0))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -1.7e-5)
		tmp = t_0;
	elseif (im <= 0.00027)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.32e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -1.7e-5)
		tmp = t_0;
	elseif (im <= 0.00027)
		tmp = cos(re) * -im;
	elseif (im <= 1.32e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(9.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -1.7e-5], t$95$0, If[LessEqual[im, 0.00027], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.32e+154], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.35000000000000003e154 or 1.31999999999999998e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.0%

      \[\leadsto \cos re \cdot \left(\color{blue}{-3} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.0%

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

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-3 \cdot -3 - im \cdot im}{-3 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{9} - im \cdot im\right) \cdot \cos re}{-3 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(9 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(9 - im \cdot im\right) \cdot \cos re}{im + -3}} \]

   if -1.35000000000000003e154 < im < -1.7e-5 or 2.70000000000000003e-4 < im < 1.31999999999999998e154

   1. Initial program 99.8%

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

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in re around 0 77.5%

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

   if -1.7e-5 < im < 2.70000000000000003e-4

   1. Initial program 8.2%

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

      1. sub0-neg 8.2%

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

   3. Simplified 8.2%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.00027:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \]

Alternative 4: 83.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ t_1 := \frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ t_2 := \left(-3 - im\right) \cdot \mathsf{fma}\left(re, 0.5 \cdot re, 1\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -4.8 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.48 \cdot 10^{+23}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 515000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im))
        (t_1 (/ (* (cos re) (- 9.0 (* im im))) (+ im -3.0)))
        (t_2 (* (- -3.0 im) (fma re (* 0.5 re) 1.0))))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -4.8e+85)
       t_0
       (if (<= im -1.48e+23)
         t_2
         (if (<= im 515000000.0)
           (* (cos re) (- im))
           (if (<= im 4.2e+100) t_2 (if (<= im 1.32e+154) t_0 t_1))))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double t_2 = (-3.0 - im) * fma(re, (0.5 * re), 1.0);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -4.8e+85) {
		tmp = t_0;
	} else if (im <= -1.48e+23) {
		tmp = t_2;
	} else if (im <= 515000000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 4.2e+100) {
		tmp = t_2;
	} else if (im <= 1.32e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	t_1 = Float64(Float64(cos(re) * Float64(9.0 - Float64(im * im))) / Float64(im + -3.0))
	t_2 = Float64(Float64(-3.0 - im) * fma(re, Float64(0.5 * re), 1.0))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -4.8e+85)
		tmp = t_0;
	elseif (im <= -1.48e+23)
		tmp = t_2;
	elseif (im <= 515000000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 4.2e+100)
		tmp = t_2;
	elseif (im <= 1.32e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(9.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-3.0 - im), $MachinePrecision] * N[(re * N[(0.5 * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -4.8e+85], t$95$0, If[LessEqual[im, -1.48e+23], t$95$2, If[LessEqual[im, 515000000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 4.2e+100], t$95$2, If[LessEqual[im, 1.32e+154], t$95$0, t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.35000000000000003e154 or 1.31999999999999998e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.0%

      \[\leadsto \cos re \cdot \left(\color{blue}{-3} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.0%

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

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-3 \cdot -3 - im \cdot im}{-3 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{9} - im \cdot im\right) \cdot \cos re}{-3 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(9 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(9 - im \cdot im\right) \cdot \cos re}{im + -3}} \]

   if -1.35000000000000003e154 < im < -4.79999999999999993e85 or 4.1999999999999997e100 < im < 1.31999999999999998e154

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 85.4%

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

      1. mul-1-neg 85.4%

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

      2. unsub-neg 85.4%

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

      3. *-commutative 85.4%

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

      4. associate-*l* 85.4%

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

      5. distribute-lft-out-- 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in re around 0 60.8%

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

   if -4.79999999999999993e85 < im < -1.4799999999999999e23 or 5.15e8 < im < 4.1999999999999997e100

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.1%

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

      1. mul-1-neg 5.1%

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

      2. unsub-neg 5.1%

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

      3. *-commutative 5.1%

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

      4. associate-*l* 5.1%

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

      5. distribute-lft-out-- 5.1%

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

   6. Simplified 5.1%

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

   8. Applied egg-rr 3.6%

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

   9. Taylor expanded in re around 0 8.5%

      \[\leadsto \color{blue}{-1 \cdot \left(3 + im\right) + 0.5 \cdot \left({re}^{2} \cdot \left(3 + im\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 8.5%

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

       2. distribute-rgt-out 8.5%

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

       3. +-commutative 8.5%

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

       4. metadata-eval 8.5%

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

       5. metadata-eval 8.5%

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

       6. distribute-lft-neg-in 8.5%

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

       7. unpow2 8.5%

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

       8. associate-*r* 8.5%

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

       9. *-commutative 8.5%

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

       10. distribute-neg-in 8.5%

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

       11. +-commutative 8.5%

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

       12. distribute-neg-in 8.5%

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

       13. *-commutative 8.5%

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

       14. distribute-lft-neg-in 8.5%

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

       15. distribute-lft-neg-in 8.5%

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

       16. metadata-eval 8.5%

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

       17. metadata-eval 8.5%

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

   11. Simplified 8.5%

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

       1. distribute-rgt-in 8.5%

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

       2. add-sqr-sqrt 8.5%

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

       3. associate-*l* 8.5%

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

       4. *-commutative 8.5%

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

       5. fma-def 8.5%

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

   13. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot \sqrt{-0.5}, \left(re \cdot \sqrt{-0.5}\right) \cdot \left(im + 3\right), \mathsf{fma}\left(im, -1, -3\right)\right)} \]
   14. Step-by-step derivation

       1. fma-udef 0.0%

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

       2. +-commutative 0.0%

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

       3. *-lft-identity 0.0%

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

       4. associate-*r* 0.0%

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

       5. swap-sqr 0.0%

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

       6. unpow2 0.0%

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

       7. rem-square-sqrt 28.8%

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

       8. metadata-eval 28.8%

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

       9. associate-*l* 28.8%

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

       10. unpow2 28.8%

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

       11. associate-*r* 28.8%

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

       12. *-commutative 28.8%

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

       13. associate-*r* 28.8%

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

       14. *-commutative 28.8%

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

       15. *-commutative 28.8%

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

       16. associate-*l* 28.8%

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

       17. neg-mul-1 28.8%

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

       18. distribute-lft-neg-in 28.8%

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

       19. distribute-rgt-neg-in 28.8%

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

       20. neg-mul-1 28.8%

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

       21. distribute-rgt-in 28.8%

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

       22. metadata-eval 28.8%

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

       23. fma-udef 28.8%

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

   15. Simplified 28.8%

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

   if -1.4799999999999999e23 < im < 5.15e8

   1. Initial program 13.4%

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

      1. sub0-neg 13.4%

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

   3. Simplified 13.4%

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

   4. Taylor expanded in im around 0 94.7%

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

      1. mul-1-neg 94.7%

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

      2. *-commutative 94.7%

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

      3. distribute-lft-neg-in 94.7%

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

   6. Simplified 94.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{elif}\;im \leq -4.8 \cdot 10^{+85}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq -1.48 \cdot 10^{+23}:\\ \;\;\;\;\left(-3 - im\right) \cdot \mathsf{fma}\left(re, 0.5 \cdot re, 1\right)\\ \mathbf{elif}\;im \leq 515000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+100}:\\ \;\;\;\;\left(-3 - im\right) \cdot \mathsf{fma}\left(re, 0.5 \cdot re, 1\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \]

Alternative 5: 81.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ t_1 := \left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot t_0\\ \mathbf{if}\;im \leq -1.75 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\left(-3 - im\right) \cdot \mathsf{fma}\left(re, 0.5 \cdot re, 1\right)\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 9000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(t_0 \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im))
        (t_1 (* (+ (* -0.5 (* re re)) 1.0) t_0)))
   (if (<= im -1.75e+84)
     t_1
     (if (<= im -1.55e+23)
       (* (- -3.0 im) (fma re (* 0.5 re) 1.0))
       (if (<= im -1.7e-5)
         t_1
         (if (<= im 9000000000.0)
           (* (cos re) (- im))
           (if (<= im 1.32e+154)
             (* re (* t_0 (* 0.5 re)))
             (/ (* (cos re) (- 9.0 (* im im))) (+ im -3.0)))))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double t_1 = ((-0.5 * (re * re)) + 1.0) * t_0;
	double tmp;
	if (im <= -1.75e+84) {
		tmp = t_1;
	} else if (im <= -1.55e+23) {
		tmp = (-3.0 - im) * fma(re, (0.5 * re), 1.0);
	} else if (im <= -1.7e-5) {
		tmp = t_1;
	} else if (im <= 9000000000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 1.32e+154) {
		tmp = re * (t_0 * (0.5 * re));
	} else {
		tmp = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	t_1 = Float64(Float64(Float64(-0.5 * Float64(re * re)) + 1.0) * t_0)
	tmp = 0.0
	if (im <= -1.75e+84)
		tmp = t_1;
	elseif (im <= -1.55e+23)
		tmp = Float64(Float64(-3.0 - im) * fma(re, Float64(0.5 * re), 1.0));
	elseif (im <= -1.7e-5)
		tmp = t_1;
	elseif (im <= 9000000000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.32e+154)
		tmp = Float64(re * Float64(t_0 * Float64(0.5 * re)));
	else
		tmp = Float64(Float64(cos(re) * Float64(9.0 - Float64(im * im))) / Float64(im + -3.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[im, -1.75e+84], t$95$1, If[LessEqual[im, -1.55e+23], N[(N[(-3.0 - im), $MachinePrecision] * N[(re * N[(0.5 * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -1.7e-5], t$95$1, If[LessEqual[im, 9000000000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.32e+154], N[(re * N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * N[(9.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -1.7499999999999999e84 or -1.54999999999999985e23 < im < -1.7e-5

   1. Initial program 99.6%

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

      1. sub0-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in im around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. unsub-neg 86.8%

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

      3. *-commutative 86.8%

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

      4. associate-*l* 86.8%

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

      5. distribute-lft-out-- 86.8%

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

   6. Simplified 86.8%

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

   7. Taylor expanded in re around 0 10.3%

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

      1. associate--l+ 10.3%

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

      2. associate-*r* 10.3%

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

      3. distribute-lft1-in 82.3%

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

      4. unpow2 82.3%

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

   9. Simplified 82.3%

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

   if -1.7499999999999999e84 < im < -1.54999999999999985e23

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.9%

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

      1. mul-1-neg 4.9%

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

      2. unsub-neg 4.9%

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

      3. *-commutative 4.9%

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

      4. associate-*l* 4.9%

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

      5. distribute-lft-out-- 4.9%

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

   6. Simplified 4.9%

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

   8. Applied egg-rr 3.5%

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

   9. Taylor expanded in re around 0 2.5%

      \[\leadsto \color{blue}{-1 \cdot \left(3 + im\right) + 0.5 \cdot \left({re}^{2} \cdot \left(3 + im\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 2.5%

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

       2. distribute-rgt-out 2.5%

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

       3. +-commutative 2.5%

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

       4. metadata-eval 2.5%

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

       5. metadata-eval 2.5%

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

       6. distribute-lft-neg-in 2.5%

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

       7. unpow2 2.5%

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

       8. associate-*r* 2.5%

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

       9. *-commutative 2.5%

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

       10. distribute-neg-in 2.5%

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

       11. +-commutative 2.5%

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

       12. distribute-neg-in 2.5%

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

       13. *-commutative 2.5%

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

       14. distribute-lft-neg-in 2.5%

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

       15. distribute-lft-neg-in 2.5%

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

       16. metadata-eval 2.5%

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

       17. metadata-eval 2.5%

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

   11. Simplified 2.5%

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

       1. distribute-rgt-in 2.5%

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

       2. add-sqr-sqrt 2.5%

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

       3. associate-*l* 2.5%

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

       4. *-commutative 2.5%

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

       5. fma-def 2.5%

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

   13. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot \sqrt{-0.5}, \left(re \cdot \sqrt{-0.5}\right) \cdot \left(im + 3\right), \mathsf{fma}\left(im, -1, -3\right)\right)} \]
   14. Step-by-step derivation

       1. fma-udef 0.0%

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

       2. +-commutative 0.0%

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

       3. *-lft-identity 0.0%

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

       4. associate-*r* 0.0%

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

       5. swap-sqr 0.0%

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

       6. unpow2 0.0%

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

       7. rem-square-sqrt 29.0%

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

       8. metadata-eval 29.0%

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

       9. associate-*l* 29.0%

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

       10. unpow2 29.0%

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

       11. associate-*r* 29.0%

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

       12. *-commutative 29.0%

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

       13. associate-*r* 29.0%

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

       14. *-commutative 29.0%

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

       15. *-commutative 29.0%

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

       16. associate-*l* 29.0%

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

       17. neg-mul-1 29.0%

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

       18. distribute-lft-neg-in 29.0%

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

       19. distribute-rgt-neg-in 29.0%

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

       20. neg-mul-1 29.0%

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

       21. distribute-rgt-in 29.0%

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

       22. metadata-eval 29.0%

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

       23. fma-udef 29.0%

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

   15. Simplified 29.0%

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

   if -1.7e-5 < im < 9e9

   1. Initial program 10.6%

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

      1. sub0-neg 10.6%

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

   3. Simplified 10.6%

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

   4. Taylor expanded in im around 0 97.3%

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

      1. mul-1-neg 97.3%

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

      2. *-commutative 97.3%

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

      3. distribute-lft-neg-in 97.3%

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

   6. Simplified 97.3%

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

   if 9e9 < im < 1.31999999999999998e154

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 29.1%

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

      1. mul-1-neg 29.1%

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

      2. unsub-neg 29.1%

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

      3. *-commutative 29.1%

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

      4. associate-*l* 29.1%

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

      5. distribute-lft-out-- 29.1%

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

   6. Simplified 29.1%

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

   7. Taylor expanded in re around 0 16.5%

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

      1. associate--l+ 16.5%

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

      2. associate-*r* 16.5%

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

      3. distribute-lft1-in 34.0%

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

      4. unpow2 34.0%

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

   9. Simplified 34.0%

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

   10. Taylor expanded in re around inf 20.5%

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

       1. fma-neg 20.5%

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

       2. associate-*r* 20.5%

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

       3. unpow2 20.5%

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

       4. fma-neg 20.5%

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

   12. Simplified 20.5%

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

       1. sub-neg 20.5%

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

       2. distribute-lft-in 20.5%

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

       3. add-sqr-sqrt 0.2%

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

       4. sqrt-unprod 18.3%

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

       5. swap-sqr 18.3%

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

       6. metadata-eval 18.3%

          \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)} \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right) + \left(-0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(-im\right) \]

       7. metadata-eval 18.3%

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

       8. swap-sqr 18.3%

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

       9. associate-*l* 18.3%

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

       10. associate-*l* 18.3%

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

       11. sqrt-unprod 16.1%

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

       12. add-sqr-sqrt 16.1%

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

       13. *-commutative 16.1%

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

       14. *-commutative 16.1%

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

   14. Applied egg-rr 38.6%

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

       1. distribute-lft-out 38.6%

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

       2. fma-udef 38.6%

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

       3. associate-*l* 46.2%

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

       4. fma-neg 46.2%

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

   16. Simplified 46.2%

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

   if 1.31999999999999998e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.1%

      \[\leadsto \cos re \cdot \left(\color{blue}{-3} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.1%

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

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-3 \cdot -3 - im \cdot im}{-3 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{9} - im \cdot im\right) \cdot \cos re}{-3 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(9 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(9 - im \cdot im\right) \cdot \cos re}{im + -3}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.75 \cdot 10^{+84}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\left(-3 - im\right) \cdot \mathsf{fma}\left(re, 0.5 \cdot re, 1\right)\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 9000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \]

Alternative 6: 84.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(0.5 \cdot re\right)\right)\\ t_1 := \frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.48 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 9500000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (* re (* (- (* (pow im 3.0) -0.16666666666666666) im) (* 0.5 re))))
        (t_1 (/ (* (cos re) (- 9.0 (* im im))) (+ im -3.0))))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -1.48e+23)
       t_0
       (if (<= im 9500000000.0)
         (* (cos re) (- im))
         (if (<= im 1.32e+154) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = re * (((pow(im, 3.0) * -0.16666666666666666) - im) * (0.5 * re));
	double t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -1.48e+23) {
		tmp = t_0;
	} else if (im <= 9500000000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 1.32e+154) {
		tmp = 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 * ((((im ** 3.0d0) * (-0.16666666666666666d0)) - im) * (0.5d0 * re))
    t_1 = (cos(re) * (9.0d0 - (im * im))) / (im + (-3.0d0))
    if (im <= (-1.35d+154)) then
        tmp = t_1
    else if (im <= (-1.48d+23)) then
        tmp = t_0
    else if (im <= 9500000000.0d0) then
        tmp = cos(re) * -im
    else if (im <= 1.32d+154) then
        tmp = 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 * (((Math.pow(im, 3.0) * -0.16666666666666666) - im) * (0.5 * re));
	double t_1 = (Math.cos(re) * (9.0 - (im * im))) / (im + -3.0);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -1.48e+23) {
		tmp = t_0;
	} else if (im <= 9500000000.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.32e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (((math.pow(im, 3.0) * -0.16666666666666666) - im) * (0.5 * re))
	t_1 = (math.cos(re) * (9.0 - (im * im))) / (im + -3.0)
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_1
	elif im <= -1.48e+23:
		tmp = t_0
	elif im <= 9500000000.0:
		tmp = math.cos(re) * -im
	elif im <= 1.32e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) * Float64(0.5 * re)))
	t_1 = Float64(Float64(cos(re) * Float64(9.0 - Float64(im * im))) / Float64(im + -3.0))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -1.48e+23)
		tmp = t_0;
	elseif (im <= 9500000000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.32e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * ((((im ^ 3.0) * -0.16666666666666666) - im) * (0.5 * re));
	t_1 = (cos(re) * (9.0 - (im * im))) / (im + -3.0);
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -1.48e+23)
		tmp = t_0;
	elseif (im <= 9500000000.0)
		tmp = cos(re) * -im;
	elseif (im <= 1.32e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[re], $MachinePrecision] * N[(9.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -1.48e+23], t$95$0, If[LessEqual[im, 9500000000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.32e+154], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.35000000000000003e154 or 1.31999999999999998e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 7.0%

      \[\leadsto \cos re \cdot \left(\color{blue}{-3} - im\right) \]
   9. Step-by-step derivation

      1. *-commutative 7.0%

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

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{-3 \cdot -3 - im \cdot im}{-3 + im}} \cdot \cos re \]

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \frac{\left(\color{blue}{9} - im \cdot im\right) \cdot \cos re}{-3 + im} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\left(9 - im \cdot im\right) \cdot \cos re}{\color{blue}{im + -3}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(9 - im \cdot im\right) \cdot \cos re}{im + -3}} \]

   if -1.35000000000000003e154 < im < -1.4799999999999999e23 or 9.5e9 < im < 1.31999999999999998e154

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 32.6%

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

      1. mul-1-neg 32.6%

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

      2. unsub-neg 32.6%

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

      3. *-commutative 32.6%

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

      4. associate-*l* 32.6%

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

      5. distribute-lft-out-- 32.6%

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

   6. Simplified 32.6%

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

   7. Taylor expanded in re around 0 12.8%

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

      1. associate--l+ 12.8%

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

      2. associate-*r* 12.8%

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

      3. distribute-lft1-in 34.7%

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

      4. unpow2 34.7%

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

   9. Simplified 34.7%

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

   10. Taylor expanded in re around inf 19.7%

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

       1. fma-neg 19.7%

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

       2. associate-*r* 19.7%

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

       3. unpow2 19.7%

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

       4. fma-neg 19.7%

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

   12. Simplified 19.7%

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

       1. sub-neg 19.7%

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

       2. distribute-lft-in 19.7%

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

       3. add-sqr-sqrt 0.3%

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

       4. sqrt-unprod 13.2%

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

       5. swap-sqr 13.2%

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

       6. metadata-eval 13.2%

          \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot re\right)\right)} \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right) + \left(-0.5 \cdot \left(re \cdot re\right)\right) \cdot \left(-im\right) \]

       7. metadata-eval 13.2%

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

       8. swap-sqr 13.2%

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

       9. associate-*l* 13.2%

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

       10. associate-*l* 13.2%

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

       11. sqrt-unprod 13.5%

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

       12. add-sqr-sqrt 13.5%

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

       13. *-commutative 13.5%

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

       14. *-commutative 13.5%

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

   14. Applied egg-rr 34.0%

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

       1. distribute-lft-out 34.0%

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

       2. fma-udef 34.0%

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

       3. associate-*l* 42.3%

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

       4. fma-neg 42.3%

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

   16. Simplified 42.3%

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

   if -1.4799999999999999e23 < im < 9.5e9

   1. Initial program 14.1%

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

      1. sub0-neg 14.1%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in im around 0 94.0%

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

      1. mul-1-neg 94.0%

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

      2. *-commutative 94.0%

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

      3. distribute-lft-neg-in 94.0%

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

   6. Simplified 94.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \mathbf{elif}\;im \leq -1.48 \cdot 10^{+23}:\\ \;\;\;\;re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 9500000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(0.5 \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(9 - im \cdot im\right)}{im + -3}\\ \end{array} \]

Alternative 7: 77.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ t_1 := \left(re \cdot re\right) \cdot \left({im}^{3} \cdot 0.08333333333333333\right)\\ \mathbf{if}\;im \leq -7.2 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -600:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.4 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+28}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im))
        (t_1 (* (* re re) (* (pow im 3.0) 0.08333333333333333))))
   (if (<= im -7.2e+102)
     t_0
     (if (<= im -600.0)
       t_1
       (if (<= im -1.4e-5)
         t_0
         (if (<= im 2.6e+28)
           (* (cos re) (- im))
           (if (<= im 5.3e+88) t_1 t_0)))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double t_1 = (re * re) * (pow(im, 3.0) * 0.08333333333333333);
	double tmp;
	if (im <= -7.2e+102) {
		tmp = t_0;
	} else if (im <= -600.0) {
		tmp = t_1;
	} else if (im <= -1.4e-5) {
		tmp = t_0;
	} else if (im <= 2.6e+28) {
		tmp = cos(re) * -im;
	} else if (im <= 5.3e+88) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    t_1 = (re * re) * ((im ** 3.0d0) * 0.08333333333333333d0)
    if (im <= (-7.2d+102)) then
        tmp = t_0
    else if (im <= (-600.0d0)) then
        tmp = t_1
    else if (im <= (-1.4d-5)) then
        tmp = t_0
    else if (im <= 2.6d+28) then
        tmp = cos(re) * -im
    else if (im <= 5.3d+88) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double t_1 = (re * re) * (Math.pow(im, 3.0) * 0.08333333333333333);
	double tmp;
	if (im <= -7.2e+102) {
		tmp = t_0;
	} else if (im <= -600.0) {
		tmp = t_1;
	} else if (im <= -1.4e-5) {
		tmp = t_0;
	} else if (im <= 2.6e+28) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 5.3e+88) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	t_1 = (re * re) * (math.pow(im, 3.0) * 0.08333333333333333)
	tmp = 0
	if im <= -7.2e+102:
		tmp = t_0
	elif im <= -600.0:
		tmp = t_1
	elif im <= -1.4e-5:
		tmp = t_0
	elif im <= 2.6e+28:
		tmp = math.cos(re) * -im
	elif im <= 5.3e+88:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	t_1 = Float64(Float64(re * re) * Float64((im ^ 3.0) * 0.08333333333333333))
	tmp = 0.0
	if (im <= -7.2e+102)
		tmp = t_0;
	elseif (im <= -600.0)
		tmp = t_1;
	elseif (im <= -1.4e-5)
		tmp = t_0;
	elseif (im <= 2.6e+28)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 5.3e+88)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	t_1 = (re * re) * ((im ^ 3.0) * 0.08333333333333333);
	tmp = 0.0;
	if (im <= -7.2e+102)
		tmp = t_0;
	elseif (im <= -600.0)
		tmp = t_1;
	elseif (im <= -1.4e-5)
		tmp = t_0;
	elseif (im <= 2.6e+28)
		tmp = cos(re) * -im;
	elseif (im <= 5.3e+88)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(re * re), $MachinePrecision] * N[(N[Power[im, 3.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.2e+102], t$95$0, If[LessEqual[im, -600.0], t$95$1, If[LessEqual[im, -1.4e-5], t$95$0, If[LessEqual[im, 2.6e+28], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 5.3e+88], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.2000000000000003e102 or -600 < im < -1.39999999999999998e-5 or 5.29999999999999987e88 < im

   1. Initial program 99.8%

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

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in im around 0 95.8%

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

      1. mul-1-neg 95.8%

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

      2. unsub-neg 95.8%

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

      3. *-commutative 95.8%

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

      4. associate-*l* 95.8%

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

      5. distribute-lft-out-- 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in re around 0 71.4%

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

   if -7.2000000000000003e102 < im < -600 or 2.6000000000000002e28 < im < 5.29999999999999987e88

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. unsub-neg 7.0%

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

      3. *-commutative 7.0%

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

      4. associate-*l* 7.0%

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

      5. distribute-lft-out-- 7.0%

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

   6. Simplified 7.0%

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

   7. Taylor expanded in re around 0 23.0%

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

      1. associate--l+ 23.0%

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

      2. associate-*r* 23.0%

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

      3. distribute-lft1-in 25.0%

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

      4. unpow2 25.0%

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

   9. Simplified 25.0%

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

   10. Taylor expanded in re around inf 23.1%

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

       1. fma-neg 23.1%

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

       2. associate-*r* 23.1%

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

       3. unpow2 23.1%

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

       4. fma-neg 23.1%

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

   12. Simplified 23.1%

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

   13. Taylor expanded in im around inf 23.1%

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

       1. *-commutative 23.1%

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

       2. associate-*l* 23.1%

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

       3. unpow2 23.1%

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

   15. Simplified 23.1%

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

   if -1.39999999999999998e-5 < im < 2.6000000000000002e28

   1. Initial program 13.5%

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

      1. sub0-neg 13.5%

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

   3. Simplified 13.5%

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

   4. Taylor expanded in im around 0 94.2%

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

      1. mul-1-neg 94.2%

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

      2. *-commutative 94.2%

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

      3. distribute-lft-neg-in 94.2%

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

   6. Simplified 94.2%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{+102}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq -600:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left({im}^{3} \cdot 0.08333333333333333\right)\\ \mathbf{elif}\;im \leq -1.4 \cdot 10^{-5}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+28}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{+88}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left({im}^{3} \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 8: 77.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ t_1 := \left(-3 - im\right) \cdot \mathsf{fma}\left(re, 0.5 \cdot re, 1\right)\\ \mathbf{if}\;im \leq -4.8 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.65 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 510000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im))
        (t_1 (* (- -3.0 im) (fma re (* 0.5 re) 1.0))))
   (if (<= im -4.8e+85)
     t_0
     (if (<= im -1.65e+23)
       t_1
       (if (<= im 510000000.0)
         (* (cos re) (- im))
         (if (<= im 4.2e+100) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double t_1 = (-3.0 - im) * fma(re, (0.5 * re), 1.0);
	double tmp;
	if (im <= -4.8e+85) {
		tmp = t_0;
	} else if (im <= -1.65e+23) {
		tmp = t_1;
	} else if (im <= 510000000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 4.2e+100) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	t_1 = Float64(Float64(-3.0 - im) * fma(re, Float64(0.5 * re), 1.0))
	tmp = 0.0
	if (im <= -4.8e+85)
		tmp = t_0;
	elseif (im <= -1.65e+23)
		tmp = t_1;
	elseif (im <= 510000000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 4.2e+100)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-3.0 - im), $MachinePrecision] * N[(re * N[(0.5 * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.8e+85], t$95$0, If[LessEqual[im, -1.65e+23], t$95$1, If[LessEqual[im, 510000000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 4.2e+100], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.79999999999999993e85 or 4.1999999999999997e100 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 95.8%

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

      1. mul-1-neg 95.8%

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

      2. unsub-neg 95.8%

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

      3. *-commutative 95.8%

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

      4. associate-*l* 95.8%

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

      5. distribute-lft-out-- 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in re around 0 70.0%

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

   if -4.79999999999999993e85 < im < -1.65000000000000015e23 or 5.1e8 < im < 4.1999999999999997e100

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.1%

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

      1. mul-1-neg 5.1%

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

      2. unsub-neg 5.1%

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

      3. *-commutative 5.1%

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

      4. associate-*l* 5.1%

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

      5. distribute-lft-out-- 5.1%

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

   6. Simplified 5.1%

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

   8. Applied egg-rr 3.6%

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

   9. Taylor expanded in re around 0 8.5%

      \[\leadsto \color{blue}{-1 \cdot \left(3 + im\right) + 0.5 \cdot \left({re}^{2} \cdot \left(3 + im\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 8.5%

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

       2. distribute-rgt-out 8.5%

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

       3. +-commutative 8.5%

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

       4. metadata-eval 8.5%

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

       5. metadata-eval 8.5%

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

       6. distribute-lft-neg-in 8.5%

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

       7. unpow2 8.5%

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

       8. associate-*r* 8.5%

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

       9. *-commutative 8.5%

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

       10. distribute-neg-in 8.5%

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

       11. +-commutative 8.5%

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

       12. distribute-neg-in 8.5%

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

       13. *-commutative 8.5%

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

       14. distribute-lft-neg-in 8.5%

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

       15. distribute-lft-neg-in 8.5%

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

       16. metadata-eval 8.5%

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

       17. metadata-eval 8.5%

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

   11. Simplified 8.5%

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

       1. distribute-rgt-in 8.5%

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

       2. add-sqr-sqrt 8.5%

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

       3. associate-*l* 8.5%

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

       4. *-commutative 8.5%

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

       5. fma-def 8.5%

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

   13. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot \sqrt{-0.5}, \left(re \cdot \sqrt{-0.5}\right) \cdot \left(im + 3\right), \mathsf{fma}\left(im, -1, -3\right)\right)} \]
   14. Step-by-step derivation

       1. fma-udef 0.0%

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

       2. +-commutative 0.0%

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

       3. *-lft-identity 0.0%

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

       4. associate-*r* 0.0%

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

       5. swap-sqr 0.0%

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

       6. unpow2 0.0%

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

       7. rem-square-sqrt 28.8%

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

       8. metadata-eval 28.8%

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

       9. associate-*l* 28.8%

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

       10. unpow2 28.8%

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

       11. associate-*r* 28.8%

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

       12. *-commutative 28.8%

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

       13. associate-*r* 28.8%

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

       14. *-commutative 28.8%

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

       15. *-commutative 28.8%

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

       16. associate-*l* 28.8%

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

       17. neg-mul-1 28.8%

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

       18. distribute-lft-neg-in 28.8%

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

       19. distribute-rgt-neg-in 28.8%

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

       20. neg-mul-1 28.8%

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

       21. distribute-rgt-in 28.8%

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

       22. metadata-eval 28.8%

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

       23. fma-udef 28.8%

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

   15. Simplified 28.8%

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

   if -1.65000000000000015e23 < im < 5.1e8

   1. Initial program 13.4%

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

      1. sub0-neg 13.4%

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

   3. Simplified 13.4%

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

   4. Taylor expanded in im around 0 94.7%

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

      1. mul-1-neg 94.7%

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

      2. *-commutative 94.7%

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

      3. distribute-lft-neg-in 94.7%

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

   6. Simplified 94.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.8 \cdot 10^{+85}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq -1.65 \cdot 10^{+23}:\\ \;\;\;\;\left(-3 - im\right) \cdot \mathsf{fma}\left(re, 0.5 \cdot re, 1\right)\\ \mathbf{elif}\;im \leq 510000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+100}:\\ \;\;\;\;\left(-3 - im\right) \cdot \mathsf{fma}\left(re, 0.5 \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 9: 76.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 30000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+100}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -1.7e-5)
     t_0
     (if (<= im 30000000000.0)
       (* (cos re) (- im))
       (if (<= im 1.65e+100) (* 27.0 (+ 0.5 (* re (* re -0.25)))) t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -1.7e-5) {
		tmp = t_0;
	} else if (im <= 30000000000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 1.65e+100) {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-1.7d-5)) then
        tmp = t_0
    else if (im <= 30000000000.0d0) then
        tmp = cos(re) * -im
    else if (im <= 1.65d+100) then
        tmp = 27.0d0 * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -1.7e-5) {
		tmp = t_0;
	} else if (im <= 30000000000.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.65e+100) {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -1.7e-5:
		tmp = t_0
	elif im <= 30000000000.0:
		tmp = math.cos(re) * -im
	elif im <= 1.65e+100:
		tmp = 27.0 * (0.5 + (re * (re * -0.25)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -1.7e-5)
		tmp = t_0;
	elseif (im <= 30000000000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.65e+100)
		tmp = Float64(27.0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -1.7e-5)
		tmp = t_0;
	elseif (im <= 30000000000.0)
		tmp = cos(re) * -im;
	elseif (im <= 1.65e+100)
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -1.7e-5], t$95$0, If[LessEqual[im, 30000000000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.65e+100], N[(27.0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.7e-5 or 1.6500000000000001e100 < im

   1. Initial program 99.8%

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

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in im around 0 76.8%

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

      1. mul-1-neg 76.8%

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

      2. unsub-neg 76.8%

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

      3. *-commutative 76.8%

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

      4. associate-*l* 76.8%

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

      5. distribute-lft-out-- 76.8%

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

   6. Simplified 76.8%

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

   7. Taylor expanded in re around 0 56.5%

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

   if -1.7e-5 < im < 3e10

   1. Initial program 10.6%

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

      1. sub0-neg 10.6%

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

   3. Simplified 10.6%

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

   4. Taylor expanded in im around 0 97.3%

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

      1. mul-1-neg 97.3%

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

      2. *-commutative 97.3%

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

      3. distribute-lft-neg-in 97.3%

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

   6. Simplified 97.3%

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

   if 3e10 < im < 1.6500000000000001e100

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. distribute-rgt-out 62.1%

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

      5. *-commutative 62.1%

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

      6. unpow2 62.1%

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

      7. associate-*l* 62.1%

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

   6. Simplified 62.1%

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

   8. Applied egg-rr 22.3%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq 30000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+100}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 10: 59.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{if}\;im \leq -430:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 10600000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+231}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* im (* 0.5 (* re re))) im)))
   (if (<= im -430.0)
     t_0
     (if (<= im 10600000000000.0)
       (* (cos re) (- im))
       (if (<= im 4.5e+231) (* 27.0 (+ 0.5 (* re (* re -0.25)))) t_0)))))

C

double code(double re, double im) {
	double t_0 = (im * (0.5 * (re * re))) - im;
	double tmp;
	if (im <= -430.0) {
		tmp = t_0;
	} else if (im <= 10600000000000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 4.5e+231) {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im * (0.5d0 * (re * re))) - im
    if (im <= (-430.0d0)) then
        tmp = t_0
    else if (im <= 10600000000000.0d0) then
        tmp = cos(re) * -im
    else if (im <= 4.5d+231) then
        tmp = 27.0d0 * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * (0.5 * (re * re))) - im;
	double tmp;
	if (im <= -430.0) {
		tmp = t_0;
	} else if (im <= 10600000000000.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 4.5e+231) {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * (0.5 * (re * re))) - im
	tmp = 0
	if im <= -430.0:
		tmp = t_0
	elif im <= 10600000000000.0:
		tmp = math.cos(re) * -im
	elif im <= 4.5e+231:
		tmp = 27.0 * (0.5 + (re * (re * -0.25)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im)
	tmp = 0.0
	if (im <= -430.0)
		tmp = t_0;
	elseif (im <= 10600000000000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 4.5e+231)
		tmp = Float64(27.0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * (0.5 * (re * re))) - im;
	tmp = 0.0;
	if (im <= -430.0)
		tmp = t_0;
	elseif (im <= 10600000000000.0)
		tmp = cos(re) * -im;
	elseif (im <= 4.5e+231)
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -430.0], t$95$0, If[LessEqual[im, 10600000000000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 4.5e+231], N[(27.0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -430 or 4.49999999999999991e231 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.9%

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

      1. mul-1-neg 5.9%

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

      2. *-commutative 5.9%

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

      3. distribute-lft-neg-in 5.9%

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

   6. Simplified 5.9%

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

   7. Taylor expanded in re around 0 27.5%

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

      1. neg-mul-1 27.5%

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

      2. +-commutative 27.5%

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

      3. unsub-neg 27.5%

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

      4. *-commutative 27.5%

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

      5. *-commutative 27.5%

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

      6. associate-*l* 27.5%

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

      7. unpow2 27.5%

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

   9. Simplified 27.5%

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

   if -430 < im < 1.06e13

   1. Initial program 11.2%

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

      1. sub0-neg 11.2%

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

   3. Simplified 11.2%

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

   4. Taylor expanded in im around 0 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-lft-neg-in 97.1%

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

   6. Simplified 97.1%

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

   if 1.06e13 < im < 4.49999999999999991e231

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. distribute-rgt-out 63.0%

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

      5. *-commutative 63.0%

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

      6. unpow2 63.0%

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

      7. associate-*l* 63.0%

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

   6. Simplified 63.0%

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

   8. Applied egg-rr 21.8%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -430:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq 10600000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+231}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \end{array} \]

Alternative 11: 36.7% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \mathbf{if}\;im \leq -410:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 12500000000:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+255}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re re) (* im 0.5))))
   (if (<= im -410.0)
     t_0
     (if (<= im 12500000000.0)
       (- im)
       (if (<= im 2.6e+255) (* 27.0 (+ 0.5 (* re (* re -0.25)))) t_0)))))

C

double code(double re, double im) {
	double t_0 = (re * re) * (im * 0.5);
	double tmp;
	if (im <= -410.0) {
		tmp = t_0;
	} else if (im <= 12500000000.0) {
		tmp = -im;
	} else if (im <= 2.6e+255) {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re * re) * (im * 0.5d0)
    if (im <= (-410.0d0)) then
        tmp = t_0
    else if (im <= 12500000000.0d0) then
        tmp = -im
    else if (im <= 2.6d+255) then
        tmp = 27.0d0 * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) * (im * 0.5);
	double tmp;
	if (im <= -410.0) {
		tmp = t_0;
	} else if (im <= 12500000000.0) {
		tmp = -im;
	} else if (im <= 2.6e+255) {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) * (im * 0.5)
	tmp = 0
	if im <= -410.0:
		tmp = t_0
	elif im <= 12500000000.0:
		tmp = -im
	elif im <= 2.6e+255:
		tmp = 27.0 * (0.5 + (re * (re * -0.25)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) * Float64(im * 0.5))
	tmp = 0.0
	if (im <= -410.0)
		tmp = t_0;
	elseif (im <= 12500000000.0)
		tmp = Float64(-im);
	elseif (im <= 2.6e+255)
		tmp = Float64(27.0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) * (im * 0.5);
	tmp = 0.0;
	if (im <= -410.0)
		tmp = t_0;
	elseif (im <= 12500000000.0)
		tmp = -im;
	elseif (im <= 2.6e+255)
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -410.0], t$95$0, If[LessEqual[im, 12500000000.0], (-im), If[LessEqual[im, 2.6e+255], N[(27.0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -410 or 2.6000000000000001e255 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 69.9%

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

      1. mul-1-neg 69.9%

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

      2. unsub-neg 69.9%

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

      3. *-commutative 69.9%

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

      4. associate-*l* 69.9%

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

      5. distribute-lft-out-- 69.9%

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

   6. Simplified 69.9%

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

   7. Taylor expanded in re around 0 5.8%

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

      1. associate--l+ 5.8%

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

      2. associate-*r* 5.8%

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

      3. distribute-lft1-in 66.8%

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

      4. unpow2 66.8%

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

   9. Simplified 66.8%

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

   10. Taylor expanded in re around inf 28.3%

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

       1. fma-neg 28.3%

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

       2. associate-*r* 28.3%

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

       3. unpow2 28.3%

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

       4. fma-neg 28.3%

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

   12. Simplified 28.3%

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

   13. Taylor expanded in im around 0 25.3%

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

       1. associate-*r* 25.3%

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

       2. *-commutative 25.3%

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

       3. associate-*l* 25.3%

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

       4. unpow2 25.3%

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

   15. Simplified 25.3%

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

   if -410 < im < 1.25e10

   1. Initial program 11.2%

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

      1. sub0-neg 11.2%

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

   3. Simplified 11.2%

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

   4. Taylor expanded in im around 0 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-lft-neg-in 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in re around 0 48.1%

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

      1. neg-mul-1 48.1%

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

   9. Simplified 48.1%

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

   if 1.25e10 < im < 2.6000000000000001e255

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. distribute-rgt-out 62.5%

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

      5. *-commutative 62.5%

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

      6. unpow2 62.5%

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

      7. associate-*l* 62.5%

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

   6. Simplified 62.5%

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

   8. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -410:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 12500000000:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+255}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 37.4% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -900 \lor \neg \left(im \leq 1.35 \cdot 10^{+28}\right):\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -900.0) (not (<= im 1.35e+28)))
   (* (* re re) (* im 0.5))
   (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -900.0) || !(im <= 1.35e+28)) {
		tmp = (re * re) * (im * 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 <= (-900.0d0)) .or. (.not. (im <= 1.35d+28))) then
        tmp = (re * re) * (im * 0.5d0)
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -900.0) || !(im <= 1.35e+28)) {
		tmp = (re * re) * (im * 0.5);
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -900.0) or not (im <= 1.35e+28):
		tmp = (re * re) * (im * 0.5)
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -900.0) || !(im <= 1.35e+28))
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -900.0) || ~((im <= 1.35e+28)))
		tmp = (re * re) * (im * 0.5);
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -900.0], N[Not[LessEqual[im, 1.35e+28]], $MachinePrecision]], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -900 or 1.3500000000000001e28 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 63.3%

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

      1. mul-1-neg 63.3%

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

      2. unsub-neg 63.3%

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

      3. *-commutative 63.3%

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

      4. associate-*l* 63.3%

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

      5. distribute-lft-out-- 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in re around 0 8.4%

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

      1. associate--l+ 8.4%

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

      2. associate-*r* 8.4%

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

      3. distribute-lft1-in 58.5%

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

      4. unpow2 58.5%

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

   9. Simplified 58.5%

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

   10. Taylor expanded in re around inf 24.1%

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

       1. fma-neg 24.1%

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

       2. associate-*r* 24.1%

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

       3. unpow2 24.1%

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

       4. fma-neg 24.1%

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

   12. Simplified 24.1%

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

   13. Taylor expanded in im around 0 19.7%

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

       1. associate-*r* 19.7%

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

       2. *-commutative 19.7%

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

       3. associate-*l* 19.7%

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

       4. unpow2 19.7%

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

   15. Simplified 19.7%

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

   if -900 < im < 1.3500000000000001e28

   1. Initial program 14.1%

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

      1. sub0-neg 14.1%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in im around 0 94.0%

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

      1. mul-1-neg 94.0%

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

      2. *-commutative 94.0%

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

      3. distribute-lft-neg-in 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in re around 0 46.7%

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

      1. neg-mul-1 46.7%

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

   9. Simplified 46.7%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -900 \lor \neg \left(im \leq 1.35 \cdot 10^{+28}\right):\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 13: 36.5% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+167}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.5e+167)
   (* 27.0 (+ 0.5 (* re (* re -0.25))))
   (- (* im (* 0.5 (* re re))) im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.5e+167) {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = (im * (0.5 * (re * 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 (re <= (-2.5d+167)) then
        tmp = 27.0d0 * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = (im * (0.5d0 * (re * re))) - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.5e+167) {
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = (im * (0.5 * (re * re))) - im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.5e+167:
		tmp = 27.0 * (0.5 + (re * (re * -0.25)))
	else:
		tmp = (im * (0.5 * (re * re))) - im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.5e+167)
		tmp = Float64(27.0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.5e+167)
		tmp = 27.0 * (0.5 + (re * (re * -0.25)));
	else
		tmp = (im * (0.5 * (re * re))) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.5e+167], N[(27.0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.4999999999999998e167

   1. Initial program 50.1%

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

      1. sub0-neg 50.1%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. distribute-rgt-out 18.2%

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

      5. *-commutative 18.2%

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

      6. unpow2 18.2%

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

      7. associate-*l* 18.2%

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

   6. Simplified 18.2%

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

   8. Applied egg-rr 31.4%

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

   if -2.4999999999999998e167 < re

   1. Initial program 60.4%

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

      1. sub0-neg 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in im around 0 46.2%

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

      1. mul-1-neg 46.2%

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

      2. *-commutative 46.2%

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

      3. distribute-lft-neg-in 46.2%

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

   6. Simplified 46.2%

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

   7. Taylor expanded in re around 0 33.3%

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

      1. neg-mul-1 33.3%

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

      2. +-commutative 33.3%

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

      3. unsub-neg 33.3%

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

      4. *-commutative 33.3%

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

      5. *-commutative 33.3%

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

      6. associate-*l* 33.3%

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

      7. unpow2 33.3%

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

   9. Simplified 33.3%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+167}:\\ \;\;\;\;27 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \end{array} \]

Alternative 14: 32.1% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 8 \cdot 10^{+153}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 8e+153) (- im) (* (* re re) 0.75)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 8e+153) {
		tmp = -im;
	} else {
		tmp = (re * re) * 0.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 8d+153) then
        tmp = -im
    else
        tmp = (re * re) * 0.75d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 8e+153) {
		tmp = -im;
	} else {
		tmp = (re * re) * 0.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 8e+153:
		tmp = -im
	else:
		tmp = (re * re) * 0.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 8e+153)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(re * re) * 0.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 8e+153)
		tmp = -im;
	else
		tmp = (re * re) * 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 8e+153], (-im), N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8e153

   1. Initial program 58.0%

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

      1. sub0-neg 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in im around 0 48.9%

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

      1. mul-1-neg 48.9%

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

      2. *-commutative 48.9%

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

      3. distribute-lft-neg-in 48.9%

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

   6. Simplified 48.9%

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

   7. Taylor expanded in re around 0 26.7%

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

      1. neg-mul-1 26.7%

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

   9. Simplified 26.7%

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

   if 8e153 < re

   1. Initial program 68.2%

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

      1. sub0-neg 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. distribute-rgt-out 33.3%

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

      5. *-commutative 33.3%

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

      6. unpow2 33.3%

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

      7. associate-*l* 33.3%

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

   6. Simplified 33.3%

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

   8. Applied egg-rr 33.9%

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

   9. Taylor expanded in re around inf 33.9%

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

       1. *-commutative 33.9%

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

       2. unpow2 33.9%

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

       3. associate-*l* 33.9%

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

   11. Simplified 33.9%

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

   12. Taylor expanded in re around 0 33.9%

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

       1. unpow2 33.9%

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

   14. Simplified 33.9%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8 \cdot 10^{+153}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \end{array} \]

Alternative 15: 30.0% 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 59.1%

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

   1. sub0-neg 59.1%

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

3. Simplified 59.1%

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

4. Taylor expanded in im around 0 47.5%

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

   1. mul-1-neg 47.5%

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

   2. *-commutative 47.5%

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

   3. distribute-lft-neg-in 47.5%

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

6. Simplified 47.5%

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

7. Taylor expanded in re around 0 24.3%

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

   1. neg-mul-1 24.3%

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

9. Simplified 24.3%

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

10. Final simplification 24.3%

    \[\leadsto -im \]

Alternative 16: 2.9% accurate, 309.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -1.5

Julia

function code(re, im)
	return -1.5
end

MATLAB

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

Wolfram

code[re_, im_] := -1.5

Derivation

1. Initial program 59.1%

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

   1. sub0-neg 59.1%

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

3. Simplified 59.1%

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

4. Taylor expanded in re around 0 2.7%

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

   1. +-commutative 2.7%

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

   2. *-commutative 2.7%

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

   3. associate-*r* 2.7%

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

   4. distribute-rgt-out 44.9%

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

   5. *-commutative 44.9%

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

   6. unpow2 44.9%

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

   7. associate-*l* 44.9%

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

6. Simplified 44.9%

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

8. Applied egg-rr 7.9%

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

9. Taylor expanded in re around 0 2.7%

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

10. Final simplification 2.7%

    \[\leadsto -1.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023195 
(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))))