math.cos on complex, imaginary part

Percentage Accurate: 65.6% → 99.8%

Time: 8.5s

Alternatives: 14

Speedup: 2.8×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.2 \lor \neg \left(t_0 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.2) (not (<= t_0 2e-6)))
     (* t_0 (* 0.5 (sin re)))
     (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 2e-6)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.2d0)) .or. (.not. (t_0 <= 2d-6))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 2e-6)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.2) or not (t_0 <= 2e-6):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.2) || !(t_0 <= 2e-6))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.2) || ~((t_0 <= 2e-6)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.2], N[Not[LessEqual[t$95$0, 2e-6]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[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 (neg.f64 im)) (exp.f64 im)) < -0.20000000000000001 or 1.99999999999999991e-6 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   if -0.20000000000000001 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 1.99999999999999991e-6

   1. Initial program 28.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out-- 99.9%

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

      6. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.2 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 95.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+103} \lor \neg \left(im \leq -1000\right) \land \left(im \leq 0.205 \lor \neg \left(im \leq 5.5 \cdot 10^{+102}\right)\right):\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -5e+103)
         (and (not (<= im -1000.0)) (or (<= im 0.205) (not (<= im 5.5e+102)))))
   (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
   (* (- (exp (- im)) (exp im)) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -5e+103) || (!(im <= -1000.0) && ((im <= 0.205) || !(im <= 5.5e+102)))) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = (exp(-im) - exp(im)) * (0.5 * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-5d+103)) .or. (.not. (im <= (-1000.0d0))) .and. (im <= 0.205d0) .or. (.not. (im <= 5.5d+102))) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = (exp(-im) - exp(im)) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -5e+103) || (!(im <= -1000.0) && ((im <= 0.205) || !(im <= 5.5e+102)))) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = (Math.exp(-im) - Math.exp(im)) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -5e+103) or (not (im <= -1000.0) and ((im <= 0.205) or not (im <= 5.5e+102))):
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = (math.exp(-im) - math.exp(im)) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -5e+103) || (!(im <= -1000.0) && ((im <= 0.205) || !(im <= 5.5e+102))))
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -5e+103) || (~((im <= -1000.0)) && ((im <= 0.205) || ~((im <= 5.5e+102)))))
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = (exp(-im) - exp(im)) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -5e+103], And[N[Not[LessEqual[im, -1000.0]], $MachinePrecision], Or[LessEqual[im, 0.205], N[Not[LessEqual[im, 5.5e+102]], $MachinePrecision]]]], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -5e103 or -1e3 < im < 0.204999999999999988 or 5.49999999999999981e102 < im

   1. Initial program 55.4%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. mul-1-neg 99.2%

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

      3. unsub-neg 99.2%

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

      4. associate-*r* 99.2%

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

      5. distribute-rgt-out-- 99.2%

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

      6. *-commutative 99.2%

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

   4. Simplified 99.2%

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

   if -5e103 < im < -1e3 or 0.204999999999999988 < im < 5.49999999999999981e102

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0 85.5%

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

      1. associate-*r* 85.5%

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

      2. *-commutative 85.5%

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

   4. Simplified 85.5%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+103} \lor \neg \left(im \leq -1000\right) \land \left(im \leq 0.205 \lor \neg \left(im \leq 5.5 \cdot 10^{+102}\right)\right):\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 3: 87.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -5 \cdot 10^{+103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -68000000000000:\\ \;\;\;\;re \cdot \sqrt{0.027777777777777776 \cdot {im}^{6}}\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+20} \lor \neg \left(im \leq 5.5 \cdot 10^{+102}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot {re}^{3} - re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))))
   (if (<= im -5e+103)
     t_0
     (if (<= im -68000000000000.0)
       (* re (sqrt (* 0.027777777777777776 (pow im 6.0))))
       (if (or (<= im 5.4e+20) (not (<= im 5.5e+102)))
         t_0
         (* im (- (* -0.16666666666666666 (pow re 3.0)) re)))))))

C

double code(double re, double im) {
	double t_0 = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -5e+103) {
		tmp = t_0;
	} else if (im <= -68000000000000.0) {
		tmp = re * sqrt((0.027777777777777776 * pow(im, 6.0)));
	} else if ((im <= 5.4e+20) || !(im <= 5.5e+102)) {
		tmp = t_0;
	} else {
		tmp = im * ((-0.16666666666666666 * pow(re, 3.0)) - re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    if (im <= (-5d+103)) then
        tmp = t_0
    else if (im <= (-68000000000000.0d0)) then
        tmp = re * sqrt((0.027777777777777776d0 * (im ** 6.0d0)))
    else if ((im <= 5.4d+20) .or. (.not. (im <= 5.5d+102))) then
        tmp = t_0
    else
        tmp = im * (((-0.16666666666666666d0) * (re ** 3.0d0)) - re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -5e+103) {
		tmp = t_0;
	} else if (im <= -68000000000000.0) {
		tmp = re * Math.sqrt((0.027777777777777776 * Math.pow(im, 6.0)));
	} else if ((im <= 5.4e+20) || !(im <= 5.5e+102)) {
		tmp = t_0;
	} else {
		tmp = im * ((-0.16666666666666666 * Math.pow(re, 3.0)) - re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	tmp = 0
	if im <= -5e+103:
		tmp = t_0
	elif im <= -68000000000000.0:
		tmp = re * math.sqrt((0.027777777777777776 * math.pow(im, 6.0)))
	elif (im <= 5.4e+20) or not (im <= 5.5e+102):
		tmp = t_0
	else:
		tmp = im * ((-0.16666666666666666 * math.pow(re, 3.0)) - re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	tmp = 0.0
	if (im <= -5e+103)
		tmp = t_0;
	elseif (im <= -68000000000000.0)
		tmp = Float64(re * sqrt(Float64(0.027777777777777776 * (im ^ 6.0))));
	elseif ((im <= 5.4e+20) || !(im <= 5.5e+102))
		tmp = t_0;
	else
		tmp = Float64(im * Float64(Float64(-0.16666666666666666 * (re ^ 3.0)) - re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	tmp = 0.0;
	if (im <= -5e+103)
		tmp = t_0;
	elseif (im <= -68000000000000.0)
		tmp = re * sqrt((0.027777777777777776 * (im ^ 6.0)));
	elseif ((im <= 5.4e+20) || ~((im <= 5.5e+102)))
		tmp = t_0;
	else
		tmp = im * ((-0.16666666666666666 * (re ^ 3.0)) - re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5e+103], t$95$0, If[LessEqual[im, -68000000000000.0], N[(re * N[Sqrt[N[(0.027777777777777776 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 5.4e+20], N[Not[LessEqual[im, 5.5e+102]], $MachinePrecision]], t$95$0, N[(im * N[(N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5e103 or -6.8e13 < im < 5.4e20 or 5.49999999999999981e102 < im

   1. Initial program 57.1%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 95.5%

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

      1. +-commutative 95.5%

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

      2. mul-1-neg 95.5%

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

      3. unsub-neg 95.5%

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

      4. associate-*r* 95.5%

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

      5. distribute-rgt-out-- 95.5%

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

      6. *-commutative 95.5%

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

   4. Simplified 95.5%

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

   if -5e103 < im < -6.8e13

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 9.4%

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

      1. +-commutative 9.4%

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

      2. mul-1-neg 9.4%

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

      3. unsub-neg 9.4%

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

      4. associate-*r* 9.4%

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

      5. distribute-rgt-out-- 9.4%

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

      6. *-commutative 9.4%

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

   4. Simplified 9.4%

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

   5. Taylor expanded in re around 0 38.8%

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

   6. Taylor expanded in im around inf 38.8%

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

      1. associate-*r* 38.8%

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

      2. *-commutative 38.8%

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

   8. Simplified 38.8%

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

      1. add-sqr-sqrt 38.8%

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

      2. sqrt-unprod 69.0%

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

      3. swap-sqr 69.0%

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

      4. metadata-eval 69.0%

         \[\leadsto re \cdot \sqrt{\color{blue}{0.027777777777777776} \cdot \left({im}^{3} \cdot {im}^{3}\right)} \]

      5. pow-prod-up 69.0%

         \[\leadsto re \cdot \sqrt{0.027777777777777776 \cdot \color{blue}{{im}^{\left(3 + 3\right)}}} \]

      6. metadata-eval 69.0%

         \[\leadsto re \cdot \sqrt{0.027777777777777776 \cdot {im}^{\color{blue}{6}}} \]

   10. Applied egg-rr 69.0%

       \[\leadsto re \cdot \color{blue}{\sqrt{0.027777777777777776 \cdot {im}^{6}}} \]

   if 5.4e20 < im < 5.49999999999999981e102

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 3.3%

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

      1. associate-*r* 3.3%

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

      2. neg-mul-1 3.3%

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

   4. Simplified 3.3%

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

   5. Taylor expanded in re around 0 1.8%

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

      1. +-commutative 1.8%

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

      2. mul-1-neg 1.8%

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

      3. unsub-neg 1.8%

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

      4. *-commutative 1.8%

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

      5. associate-*r* 1.8%

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

      6. *-commutative 1.8%

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

      7. distribute-rgt-out-- 5.8%

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

   7. Simplified 5.8%

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

      1. add-log-exp 28.4%

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

      2. *-un-lft-identity 28.4%

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

      3. log-prod 28.4%

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

      4. metadata-eval 28.4%

         \[\leadsto \color{blue}{0} + \log \left(e^{im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)}\right) \]

      5. add-log-exp 5.8%

         \[\leadsto 0 + \color{blue}{im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)} \]

      6. add-sqr-sqrt 1.4%

         \[\leadsto 0 + im \cdot \left(\color{blue}{\sqrt{0.16666666666666666 \cdot {re}^{3}} \cdot \sqrt{0.16666666666666666 \cdot {re}^{3}}} - re\right) \]

      7. sqrt-unprod 21.1%

         \[\leadsto 0 + im \cdot \left(\color{blue}{\sqrt{\left(0.16666666666666666 \cdot {re}^{3}\right) \cdot \left(0.16666666666666666 \cdot {re}^{3}\right)}} - re\right) \]

      8. swap-sqr 21.1%

         \[\leadsto 0 + im \cdot \left(\sqrt{\color{blue}{\left(0.16666666666666666 \cdot 0.16666666666666666\right) \cdot \left({re}^{3} \cdot {re}^{3}\right)}} - re\right) \]

      9. metadata-eval 21.1%

         \[\leadsto 0 + im \cdot \left(\sqrt{\color{blue}{0.027777777777777776} \cdot \left({re}^{3} \cdot {re}^{3}\right)} - re\right) \]

      10. metadata-eval 21.1%

          \[\leadsto 0 + im \cdot \left(\sqrt{\color{blue}{\left(-0.16666666666666666 \cdot -0.16666666666666666\right)} \cdot \left({re}^{3} \cdot {re}^{3}\right)} - re\right) \]

      11. swap-sqr 21.1%

          \[\leadsto 0 + im \cdot \left(\sqrt{\color{blue}{\left(-0.16666666666666666 \cdot {re}^{3}\right) \cdot \left(-0.16666666666666666 \cdot {re}^{3}\right)}} - re\right) \]

      12. *-commutative 21.1%

          \[\leadsto 0 + im \cdot \left(\sqrt{\color{blue}{\left({re}^{3} \cdot -0.16666666666666666\right)} \cdot \left(-0.16666666666666666 \cdot {re}^{3}\right)} - re\right) \]

      13. *-commutative 21.1%

          \[\leadsto 0 + im \cdot \left(\sqrt{\left({re}^{3} \cdot -0.16666666666666666\right) \cdot \color{blue}{\left({re}^{3} \cdot -0.16666666666666666\right)}} - re\right) \]

      14. sqrt-unprod 12.9%

          \[\leadsto 0 + im \cdot \left(\color{blue}{\sqrt{{re}^{3} \cdot -0.16666666666666666} \cdot \sqrt{{re}^{3} \cdot -0.16666666666666666}} - re\right) \]

      15. add-sqr-sqrt 33.7%

          \[\leadsto 0 + im \cdot \left(\color{blue}{{re}^{3} \cdot -0.16666666666666666} - re\right) \]

      16. *-commutative 33.7%

          \[\leadsto 0 + im \cdot \left(\color{blue}{-0.16666666666666666 \cdot {re}^{3}} - re\right) \]

   9. Applied egg-rr 33.7%

      \[\leadsto \color{blue}{0 + im \cdot \left(-0.16666666666666666 \cdot {re}^{3} - re\right)} \]
   10. Step-by-step derivation

       1. +-lft-identity 33.7%

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

   11. Simplified 33.7%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+103}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -68000000000000:\\ \;\;\;\;re \cdot \sqrt{0.027777777777777776 \cdot {im}^{6}}\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+20} \lor \neg \left(im \leq 5.5 \cdot 10^{+102}\right):\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot {re}^{3} - re\right)\\ \end{array} \]

Alternative 4: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3300:\\ \;\;\;\;re \cdot \left(\sqrt{0.027777777777777776 \cdot {im}^{6}} - im\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3300.0)
   (* re (- (sqrt (* 0.027777777777777776 (pow im 6.0))) im))
   (if (<= im 3.3e+15)
     (* im (- (sin re)))
     (* re (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3300.0) {
		tmp = re * (sqrt((0.027777777777777776 * pow(im, 6.0))) - im);
	} else if (im <= 3.3e+15) {
		tmp = im * -sin(re);
	} else {
		tmp = re * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-3300.0d0)) then
        tmp = re * (sqrt((0.027777777777777776d0 * (im ** 6.0d0))) - im)
    else if (im <= 3.3d+15) then
        tmp = im * -sin(re)
    else
        tmp = re * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -3300.0) {
		tmp = re * (Math.sqrt((0.027777777777777776 * Math.pow(im, 6.0))) - im);
	} else if (im <= 3.3e+15) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = re * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -3300.0:
		tmp = re * (math.sqrt((0.027777777777777776 * math.pow(im, 6.0))) - im)
	elif im <= 3.3e+15:
		tmp = im * -math.sin(re)
	else:
		tmp = re * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3300.0)
		tmp = Float64(re * Float64(sqrt(Float64(0.027777777777777776 * (im ^ 6.0))) - im));
	elseif (im <= 3.3e+15)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = Float64(re * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -3300.0)
		tmp = re * (sqrt((0.027777777777777776 * (im ^ 6.0))) - im);
	elseif (im <= 3.3e+15)
		tmp = im * -sin(re);
	else
		tmp = re * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3300.0], N[(re * N[(N[Sqrt[N[(0.027777777777777776 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.3e+15], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(re * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -3300

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 64.1%

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

      1. +-commutative 64.1%

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

      2. mul-1-neg 64.1%

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

      3. unsub-neg 64.1%

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

      4. associate-*r* 64.1%

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

      5. distribute-rgt-out-- 64.1%

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

      6. *-commutative 64.1%

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

   4. Simplified 64.1%

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

   5. Taylor expanded in re around 0 58.3%

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

      1. add-sqr-sqrt 58.3%

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

      2. sqrt-unprod 69.2%

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

      3. swap-sqr 69.2%

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

      4. metadata-eval 69.2%

         \[\leadsto re \cdot \sqrt{\color{blue}{0.027777777777777776} \cdot \left({im}^{3} \cdot {im}^{3}\right)} \]

      5. pow-prod-up 69.2%

         \[\leadsto re \cdot \sqrt{0.027777777777777776 \cdot \color{blue}{{im}^{\left(3 + 3\right)}}} \]

      6. metadata-eval 69.2%

         \[\leadsto re \cdot \sqrt{0.027777777777777776 \cdot {im}^{\color{blue}{6}}} \]

   7. Applied egg-rr 69.2%

      \[\leadsto re \cdot \left(\color{blue}{\sqrt{0.027777777777777776 \cdot {im}^{6}}} - im\right) \]

   if -3300 < im < 3.3e15

   1. Initial program 31.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 95.5%

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

      1. associate-*r* 95.5%

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

      2. neg-mul-1 95.5%

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

   4. Simplified 95.5%

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

   if 3.3e15 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 59.8%

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

      1. +-commutative 59.8%

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

      2. mul-1-neg 59.8%

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

      3. unsub-neg 59.8%

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

      4. associate-*r* 59.8%

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

      5. distribute-rgt-out-- 59.8%

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

      6. *-commutative 59.8%

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

   4. Simplified 59.8%

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

   5. Taylor expanded in re around 0 52.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3300:\\ \;\;\;\;re \cdot \left(\sqrt{0.027777777777777776 \cdot {im}^{6}} - im\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 5: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -145000:\\ \;\;\;\;re \cdot \sqrt{0.027777777777777776 \cdot {im}^{6}}\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -145000.0)
   (* re (sqrt (* 0.027777777777777776 (pow im 6.0))))
   (if (<= im 1.6e+16)
     (* im (- (sin re)))
     (* re (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -145000.0) {
		tmp = re * sqrt((0.027777777777777776 * pow(im, 6.0)));
	} else if (im <= 1.6e+16) {
		tmp = im * -sin(re);
	} else {
		tmp = re * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-145000.0d0)) then
        tmp = re * sqrt((0.027777777777777776d0 * (im ** 6.0d0)))
    else if (im <= 1.6d+16) then
        tmp = im * -sin(re)
    else
        tmp = re * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -145000.0) {
		tmp = re * Math.sqrt((0.027777777777777776 * Math.pow(im, 6.0)));
	} else if (im <= 1.6e+16) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = re * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -145000.0:
		tmp = re * math.sqrt((0.027777777777777776 * math.pow(im, 6.0)))
	elif im <= 1.6e+16:
		tmp = im * -math.sin(re)
	else:
		tmp = re * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -145000.0)
		tmp = Float64(re * sqrt(Float64(0.027777777777777776 * (im ^ 6.0))));
	elseif (im <= 1.6e+16)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = Float64(re * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -145000.0)
		tmp = re * sqrt((0.027777777777777776 * (im ^ 6.0)));
	elseif (im <= 1.6e+16)
		tmp = im * -sin(re);
	else
		tmp = re * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -145000.0], N[(re * N[Sqrt[N[(0.027777777777777776 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.6e+16], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(re * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -145000

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 64.1%

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

      1. +-commutative 64.1%

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

      2. mul-1-neg 64.1%

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

      3. unsub-neg 64.1%

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

      4. associate-*r* 64.1%

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

      5. distribute-rgt-out-- 64.1%

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

      6. *-commutative 64.1%

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

   4. Simplified 64.1%

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

   5. Taylor expanded in re around 0 58.3%

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

   6. Taylor expanded in im around inf 58.3%

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

      1. associate-*r* 58.3%

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

      2. *-commutative 58.3%

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

   8. Simplified 58.3%

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

      1. add-sqr-sqrt 58.3%

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

      2. sqrt-unprod 69.2%

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

      3. swap-sqr 69.2%

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

      4. metadata-eval 69.2%

         \[\leadsto re \cdot \sqrt{\color{blue}{0.027777777777777776} \cdot \left({im}^{3} \cdot {im}^{3}\right)} \]

      5. pow-prod-up 69.2%

         \[\leadsto re \cdot \sqrt{0.027777777777777776 \cdot \color{blue}{{im}^{\left(3 + 3\right)}}} \]

      6. metadata-eval 69.2%

         \[\leadsto re \cdot \sqrt{0.027777777777777776 \cdot {im}^{\color{blue}{6}}} \]

   10. Applied egg-rr 69.2%

       \[\leadsto re \cdot \color{blue}{\sqrt{0.027777777777777776 \cdot {im}^{6}}} \]

   if -145000 < im < 1.6e16

   1. Initial program 31.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 95.5%

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

      1. associate-*r* 95.5%

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

      2. neg-mul-1 95.5%

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

   4. Simplified 95.5%

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

   if 1.6e16 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 59.8%

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

      1. +-commutative 59.8%

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

      2. mul-1-neg 59.8%

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

      3. unsub-neg 59.8%

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

      4. associate-*r* 59.8%

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

      5. distribute-rgt-out-- 59.8%

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

      6. *-commutative 59.8%

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

   4. Simplified 59.8%

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

   5. Taylor expanded in re around 0 52.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -145000:\\ \;\;\;\;re \cdot \sqrt{0.027777777777777776 \cdot {im}^{6}}\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 6: 77.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3000 \lor \neg \left(im \leq 6 \cdot 10^{+14}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3000.0) (not (<= im 6e+14)))
   (* re (- (* (pow im 3.0) -0.16666666666666666) im))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3000.0) || !(im <= 6e+14)) {
		tmp = re * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = im * -sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-3000.0d0)) .or. (.not. (im <= 6d+14))) then
        tmp = re * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3000.0) || !(im <= 6e+14)) {
		tmp = re * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3000.0) or not (im <= 6e+14):
		tmp = re * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3000.0) || !(im <= 6e+14))
		tmp = Float64(re * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3000.0) || ~((im <= 6e+14)))
		tmp = re * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3000.0], N[Not[LessEqual[im, 6e+14]], $MachinePrecision]], N[(re * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3e3 or 6e14 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 61.9%

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

      1. +-commutative 61.9%

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

      2. mul-1-neg 61.9%

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

      3. unsub-neg 61.9%

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

      4. associate-*r* 61.9%

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

      5. distribute-rgt-out-- 61.9%

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

      6. *-commutative 61.9%

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

   4. Simplified 61.9%

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

   5. Taylor expanded in re around 0 55.5%

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

   if -3e3 < im < 6e14

   1. Initial program 31.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 95.5%

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

      1. associate-*r* 95.5%

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

      2. neg-mul-1 95.5%

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

   4. Simplified 95.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3000 \lor \neg \left(im \leq 6 \cdot 10^{+14}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 7: 77.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -90000 \lor \neg \left(im \leq 7 \cdot 10^{+14}\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -90000.0) (not (<= im 7e+14)))
   (* -0.16666666666666666 (* re (pow im 3.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -90000.0) || !(im <= 7e+14)) {
		tmp = -0.16666666666666666 * (re * pow(im, 3.0));
	} else {
		tmp = im * -sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-90000.0d0)) .or. (.not. (im <= 7d+14))) then
        tmp = (-0.16666666666666666d0) * (re * (im ** 3.0d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -90000.0) || !(im <= 7e+14)) {
		tmp = -0.16666666666666666 * (re * Math.pow(im, 3.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -90000.0) or not (im <= 7e+14):
		tmp = -0.16666666666666666 * (re * math.pow(im, 3.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -90000.0) || !(im <= 7e+14))
		tmp = Float64(-0.16666666666666666 * Float64(re * (im ^ 3.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -90000.0) || ~((im <= 7e+14)))
		tmp = -0.16666666666666666 * (re * (im ^ 3.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -90000.0], N[Not[LessEqual[im, 7e+14]], $MachinePrecision]], N[(-0.16666666666666666 * N[(re * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -9e4 or 7e14 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 61.9%

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

      1. +-commutative 61.9%

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

      2. mul-1-neg 61.9%

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

      3. unsub-neg 61.9%

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

      4. associate-*r* 61.9%

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

      5. distribute-rgt-out-- 61.9%

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

      6. *-commutative 61.9%

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

   4. Simplified 61.9%

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

   5. Taylor expanded in re around 0 55.5%

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

   6. Taylor expanded in im around inf 55.5%

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

   if -9e4 < im < 7e14

   1. Initial program 31.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 95.5%

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

      1. associate-*r* 95.5%

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

      2. neg-mul-1 95.5%

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

   4. Simplified 95.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -90000 \lor \neg \left(im \leq 7 \cdot 10^{+14}\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 8: 56.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.2 \cdot 10^{+14}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot -2\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.2e+14)
   (* (* 0.5 re) (* im -2.0))
   (if (<= im 6.5e+15) (* im (- (sin re))) (* im (- re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.2e+14) {
		tmp = (0.5 * re) * (im * -2.0);
	} else if (im <= 6.5e+15) {
		tmp = im * -sin(re);
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.2d+14)) then
        tmp = (0.5d0 * re) * (im * (-2.0d0))
    else if (im <= 6.5d+15) then
        tmp = im * -sin(re)
    else
        tmp = im * -re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.2e+14) {
		tmp = (0.5 * re) * (im * -2.0);
	} else if (im <= 6.5e+15) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.2e+14:
		tmp = (0.5 * re) * (im * -2.0)
	elif im <= 6.5e+15:
		tmp = im * -math.sin(re)
	else:
		tmp = im * -re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.2e+14)
		tmp = Float64(Float64(0.5 * re) * Float64(im * -2.0));
	elseif (im <= 6.5e+15)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = Float64(im * Float64(-re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.2e+14)
		tmp = (0.5 * re) * (im * -2.0);
	elseif (im <= 6.5e+15)
		tmp = im * -sin(re);
	else
		tmp = im * -re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.2e+14], N[(N[(0.5 * re), $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.5e+15], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(im * (-re)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.2e14

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0 79.7%

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

      1. associate-*r* 79.7%

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

      2. *-commutative 79.7%

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

   4. Simplified 79.7%

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

   5. Taylor expanded in im around 0 11.8%

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

   if -1.2e14 < im < 6.5e15

   1. Initial program 32.6%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 94.1%

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

      1. associate-*r* 94.1%

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

      2. neg-mul-1 94.1%

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

   4. Simplified 94.1%

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

   if 6.5e15 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 4.2%

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

      1. associate-*r* 4.2%

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

      2. neg-mul-1 4.2%

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

   4. Simplified 4.2%

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

   5. Taylor expanded in re around 0 12.0%

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

      1. associate-*r* 12.0%

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

      2. neg-mul-1 12.0%

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

   7. Simplified 12.0%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.2 \cdot 10^{+14}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot -2\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]

Alternative 9: 32.9% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in re around 0 52.4%

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

   1. associate-*r* 52.4%

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

   2. *-commutative 52.4%

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

4. Simplified 52.4%

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

5. Taylor expanded in im around 0 32.8%

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

6. Final simplification 32.8%

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

Alternative 10: 32.9% accurate, 77.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in im around 0 51.0%

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

   1. associate-*r* 51.0%

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

   2. neg-mul-1 51.0%

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

4. Simplified 51.0%

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

5. Taylor expanded in re around 0 32.4%

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

   1. associate-*r* 32.4%

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

   2. neg-mul-1 32.4%

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

7. Simplified 32.4%

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

8. Final simplification 32.4%

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

Alternative 11: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -512 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -512.0)

C

double code(double re, double im) {
	return -512.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -512.0d0
end function

Java

public static double code(double re, double im) {
	return -512.0;
}

Python

def code(re, im):
	return -512.0

Julia

function code(re, im)
	return -512.0
end

MATLAB

function tmp = code(re, im)
	tmp = -512.0;
end

Wolfram

code[re_, im_] := -512.0

Derivation

1. Initial program 65.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in im around 0 51.0%

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

   1. associate-*r* 51.0%

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

   2. neg-mul-1 51.0%

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

4. Simplified 51.0%

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

6. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{-512} \]

7. Final simplification 2.7%

   \[\leadsto -512 \]

Alternative 12: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -0.16666666666666666 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -0.16666666666666666)

C

double code(double re, double im) {
	return -0.16666666666666666;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -0.16666666666666666d0
end function

Java

public static double code(double re, double im) {
	return -0.16666666666666666;
}

Python

def code(re, im):
	return -0.16666666666666666

Julia

function code(re, im)
	return -0.16666666666666666
end

MATLAB

function tmp = code(re, im)
	tmp = -0.16666666666666666;
end

Wolfram

code[re_, im_] := -0.16666666666666666

Derivation

1. Initial program 65.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in im around 0 51.0%

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

   1. associate-*r* 51.0%

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

   2. neg-mul-1 51.0%

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

4. Simplified 51.0%

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

6. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{-0.16666666666666666} \]

7. Final simplification 2.7%

   \[\leadsto -0.16666666666666666 \]

Alternative 13: 2.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -9.92290301275212 \cdot 10^{-8} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -9.92290301275212e-8)

C

double code(double re, double im) {
	return -9.92290301275212e-8;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -9.92290301275212d-8
end function

Java

public static double code(double re, double im) {
	return -9.92290301275212e-8;
}

Python

def code(re, im):
	return -9.92290301275212e-8

Julia

function code(re, im)
	return -9.92290301275212e-8
end

MATLAB

function tmp = code(re, im)
	tmp = -9.92290301275212e-8;
end

Wolfram

code[re_, im_] := -9.92290301275212e-8

Derivation

1. Initial program 65.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in im around 0 51.0%

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

   1. associate-*r* 51.0%

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

   2. neg-mul-1 51.0%

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

4. Simplified 51.0%

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

6. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{-9.92290301275212 \cdot 10^{-8}} \]

7. Final simplification 2.7%

   \[\leadsto -9.92290301275212 \cdot 10^{-8} \]

Alternative 14: 15.2% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 65.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in im around 0 51.0%

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

   1. associate-*r* 51.0%

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

   2. neg-mul-1 51.0%

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

4. Simplified 51.0%

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

6. Applied egg-rr 13.5%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 13.5%

   \[\leadsto 0 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (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 (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-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.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(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 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[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[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023271 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))