math.cos on complex, imaginary part

Percentage Accurate: 65.4% → 99.3%

Time: 10.2s

Alternatives: 11

Speedup: 2.7×

• 49.2% 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: 65.4% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-14)))
     (* 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 <= -((double) INFINITY)) || !(t_0 <= 5e-14)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(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-14)) {
		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 <= -math.inf) or not (t_0 <= 5e-14):
		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 <= Float64(-Inf)) || !(t_0 <= 5e-14))
		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 <= -Inf) || ~((t_0 <= 5e-14)))
		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, (-Infinity)], N[Not[LessEqual[t$95$0, 5e-14]], $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)) < -inf.0 or 5.0000000000000002e-14 < (-.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 -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 5.0000000000000002e-14

   1. Initial program 28.4%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -2.7 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.066:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00095:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (- (exp (- im)) (exp im)) re)))
        (t_1 (* -0.16666666666666666 (* (sin re) (pow im 3.0)))))
   (if (<= im -2.7e+101)
     t_1
     (if (<= im -0.066)
       t_0
       (if (<= im 0.00095)
         (* im (- (sin re)))
         (if (<= im 5.5e+102) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	double t_1 = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	double tmp;
	if (im <= -2.7e+101) {
		tmp = t_1;
	} else if (im <= -0.066) {
		tmp = t_0;
	} else if (im <= 0.00095) {
		tmp = im * -sin(re);
	} else if (im <= 5.5e+102) {
		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)) * re)
    t_1 = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    if (im <= (-2.7d+101)) then
        tmp = t_1
    else if (im <= (-0.066d0)) then
        tmp = t_0
    else if (im <= 0.00095d0) then
        tmp = im * -sin(re)
    else if (im <= 5.5d+102) 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)) * re);
	double t_1 = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -2.7e+101) {
		tmp = t_1;
	} else if (im <= -0.066) {
		tmp = t_0;
	} else if (im <= 0.00095) {
		tmp = im * -Math.sin(re);
	} else if (im <= 5.5e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	t_1 = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -2.7e+101:
		tmp = t_1
	elif im <= -0.066:
		tmp = t_0
	elif im <= 0.00095:
		tmp = im * -math.sin(re)
	elif im <= 5.5e+102:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re))
	t_1 = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -2.7e+101)
		tmp = t_1;
	elseif (im <= -0.066)
		tmp = t_0;
	elseif (im <= 0.00095)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 5.5e+102)
		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)) * re);
	t_1 = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	tmp = 0.0;
	if (im <= -2.7e+101)
		tmp = t_1;
	elseif (im <= -0.066)
		tmp = t_0;
	elseif (im <= 0.00095)
		tmp = im * -sin(re);
	elseif (im <= 5.5e+102)
		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[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.7e+101], t$95$1, If[LessEqual[im, -0.066], t$95$0, If[LessEqual[im, 0.00095], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 5.5e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.70000000000000006e101 or 5.49999999999999981e102 < 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 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

      5. distribute-lft-out-- 99.0%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in im around inf 99.0%

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

   if -2.70000000000000006e101 < im < -0.066000000000000003 or 9.49999999999999998e-4 < 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 75.9%

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

   if -0.066000000000000003 < im < 9.49999999999999998e-4

   1. Initial program 28.4%

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

   2. Taylor expanded in im around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.7 \cdot 10^{+101}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -0.066:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 0.00095:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 3: 95.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -2.7 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.085:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.105:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (- (exp (- im)) (exp im)) re)))
        (t_1 (* -0.16666666666666666 (* (sin re) (pow im 3.0)))))
   (if (<= im -2.7e+101)
     t_1
     (if (<= im -0.085)
       t_0
       (if (<= im 0.105)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 5.5e+102) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	double t_1 = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	double tmp;
	if (im <= -2.7e+101) {
		tmp = t_1;
	} else if (im <= -0.085) {
		tmp = t_0;
	} else if (im <= 0.105) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 5.5e+102) {
		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)) * re)
    t_1 = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    if (im <= (-2.7d+101)) then
        tmp = t_1
    else if (im <= (-0.085d0)) then
        tmp = t_0
    else if (im <= 0.105d0) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 5.5d+102) 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)) * re);
	double t_1 = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -2.7e+101) {
		tmp = t_1;
	} else if (im <= -0.085) {
		tmp = t_0;
	} else if (im <= 0.105) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 5.5e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	t_1 = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -2.7e+101:
		tmp = t_1
	elif im <= -0.085:
		tmp = t_0
	elif im <= 0.105:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 5.5e+102:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re))
	t_1 = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -2.7e+101)
		tmp = t_1;
	elseif (im <= -0.085)
		tmp = t_0;
	elseif (im <= 0.105)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 5.5e+102)
		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)) * re);
	t_1 = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	tmp = 0.0;
	if (im <= -2.7e+101)
		tmp = t_1;
	elseif (im <= -0.085)
		tmp = t_0;
	elseif (im <= 0.105)
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 5.5e+102)
		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[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.7e+101], t$95$1, If[LessEqual[im, -0.085], t$95$0, If[LessEqual[im, 0.105], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.5e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.70000000000000006e101 or 5.49999999999999981e102 < 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 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

      5. distribute-lft-out-- 99.0%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in im around inf 99.0%

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

   if -2.70000000000000006e101 < im < -0.0850000000000000061 or 0.104999999999999996 < 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 75.9%

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

   if -0.0850000000000000061 < im < 0.104999999999999996

   1. Initial program 28.4%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.7 \cdot 10^{+101}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -0.085:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 0.105:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 4: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot re\right)}\right)\\ t_1 := -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -1.15 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -620:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 680:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (log (/ 1.0 (+ 1.0 (expm1 (* im re))))))
        (t_1 (* -0.16666666666666666 (* (sin re) (pow im 3.0)))))
   (if (<= im -1.15e+99)
     t_1
     (if (<= im -620.0)
       t_0
       (if (<= im 680.0) (* im (- (sin re))) (if (<= im 5.5e+102) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = log((1.0 / (1.0 + expm1((im * re)))));
	double t_1 = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	double tmp;
	if (im <= -1.15e+99) {
		tmp = t_1;
	} else if (im <= -620.0) {
		tmp = t_0;
	} else if (im <= 680.0) {
		tmp = im * -sin(re);
	} else if (im <= 5.5e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.log((1.0 / (1.0 + Math.expm1((im * re)))));
	double t_1 = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -1.15e+99) {
		tmp = t_1;
	} else if (im <= -620.0) {
		tmp = t_0;
	} else if (im <= 680.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 5.5e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.log((1.0 / (1.0 + math.expm1((im * re)))))
	t_1 = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -1.15e+99:
		tmp = t_1
	elif im <= -620.0:
		tmp = t_0
	elif im <= 680.0:
		tmp = im * -math.sin(re)
	elif im <= 5.5e+102:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = log(Float64(1.0 / Float64(1.0 + expm1(Float64(im * re)))))
	t_1 = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -1.15e+99)
		tmp = t_1;
	elseif (im <= -620.0)
		tmp = t_0;
	elseif (im <= 680.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 5.5e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Log[N[(1.0 / N[(1.0 + N[(Exp[N[(im * re), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.15e+99], t$95$1, If[LessEqual[im, -620.0], t$95$0, If[LessEqual[im, 680.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 5.5e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.1500000000000001e99 or 5.49999999999999981e102 < 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 97.9%

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

      1. mul-1-neg 97.9%

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

      2. unsub-neg 97.9%

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

      3. *-commutative 97.9%

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

      4. associate-*l* 97.9%

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

      5. distribute-lft-out-- 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in im around inf 97.9%

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

   if -1.1500000000000001e99 < im < -620 or 680 < 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 3.0%

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

      1. mul-1-neg 3.0%

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

      2. *-commutative 3.0%

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

      3. distribute-rgt-neg-in 3.0%

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

   4. Simplified 3.0%

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

      1. distribute-rgt-neg-out 3.0%

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

      2. add-sqr-sqrt 1.4%

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

      3. sqrt-unprod 1.8%

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

      4. sqr-neg 1.8%

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

      5. sqrt-unprod 0.4%

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

      6. add-sqr-sqrt 0.7%

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

      7. log1p-expm1-u 0.4%

         \[\leadsto -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)\right)} \]

      8. log1p-udef 0.6%

         \[\leadsto -\color{blue}{\log \left(1 + \mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)\right)} \]

      9. neg-log 0.6%

         \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)}\right)} \]

      10. add-sqr-sqrt 0.3%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \color{blue}{\left(\sqrt{-\sin re} \cdot \sqrt{-\sin re}\right)}\right)}\right) \]

      11. sqrt-unprod 27.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \color{blue}{\sqrt{\left(-\sin re\right) \cdot \left(-\sin re\right)}}\right)}\right) \]

      12. sqr-neg 27.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \sqrt{\color{blue}{\sin re \cdot \sin re}}\right)}\right) \]

      13. sqrt-unprod 26.8%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \color{blue}{\left(\sqrt{\sin re} \cdot \sqrt{\sin re}\right)}\right)}\right) \]

      14. add-sqr-sqrt 61.1%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \color{blue}{\sin re}\right)}\right) \]

   6. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \sin re\right)}\right)} \]

   7. Taylor expanded in re around 0 36.5%

      \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{re \cdot im}\right)}\right) \]

   if -620 < im < 680

   1. Initial program 28.4%

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

   2. Taylor expanded in im around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.15 \cdot 10^{+99}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -620:\\ \;\;\;\;\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot re\right)}\right)\\ \mathbf{elif}\;im \leq 680:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 5: 86.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)\\ t_1 := -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -2.75 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -36000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+41}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (log (* 0.5 (* (* re re) (* im im)))))
        (t_1 (* -0.16666666666666666 (* (sin re) (pow im 3.0)))))
   (if (<= im -2.75e+107)
     t_1
     (if (<= im -36000.0)
       t_0
       (if (<= im 2e+41) (* im (- (sin re))) (if (<= im 4.4e+101) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = log((0.5 * ((re * re) * (im * im))));
	double t_1 = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	double tmp;
	if (im <= -2.75e+107) {
		tmp = t_1;
	} else if (im <= -36000.0) {
		tmp = t_0;
	} else if (im <= 2e+41) {
		tmp = im * -sin(re);
	} else if (im <= 4.4e+101) {
		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 = log((0.5d0 * ((re * re) * (im * im))))
    t_1 = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    if (im <= (-2.75d+107)) then
        tmp = t_1
    else if (im <= (-36000.0d0)) then
        tmp = t_0
    else if (im <= 2d+41) then
        tmp = im * -sin(re)
    else if (im <= 4.4d+101) 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 = Math.log((0.5 * ((re * re) * (im * im))));
	double t_1 = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -2.75e+107) {
		tmp = t_1;
	} else if (im <= -36000.0) {
		tmp = t_0;
	} else if (im <= 2e+41) {
		tmp = im * -Math.sin(re);
	} else if (im <= 4.4e+101) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.log((0.5 * ((re * re) * (im * im))))
	t_1 = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -2.75e+107:
		tmp = t_1
	elif im <= -36000.0:
		tmp = t_0
	elif im <= 2e+41:
		tmp = im * -math.sin(re)
	elif im <= 4.4e+101:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = log(Float64(0.5 * Float64(Float64(re * re) * Float64(im * im))))
	t_1 = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -2.75e+107)
		tmp = t_1;
	elseif (im <= -36000.0)
		tmp = t_0;
	elseif (im <= 2e+41)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 4.4e+101)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = log((0.5 * ((re * re) * (im * im))));
	t_1 = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	tmp = 0.0;
	if (im <= -2.75e+107)
		tmp = t_1;
	elseif (im <= -36000.0)
		tmp = t_0;
	elseif (im <= 2e+41)
		tmp = im * -sin(re);
	elseif (im <= 4.4e+101)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Log[N[(0.5 * N[(N[(re * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.75e+107], t$95$1, If[LessEqual[im, -36000.0], t$95$0, If[LessEqual[im, 2e+41], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 4.4e+101], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.7500000000000002e107 or 4.4000000000000001e101 < 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 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

   if -2.7500000000000002e107 < im < -36000 or 2.00000000000000001e41 < im < 4.4000000000000001e101

   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 3.1%

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

      1. mul-1-neg 3.1%

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

      2. *-commutative 3.1%

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

      3. distribute-rgt-neg-in 3.1%

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

   4. Simplified 3.1%

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

      1. distribute-rgt-neg-out 3.1%

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

      2. add-sqr-sqrt 1.3%

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

      3. sqrt-unprod 1.7%

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

      4. sqr-neg 1.7%

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

      5. sqrt-unprod 0.4%

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

      6. add-sqr-sqrt 0.7%

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

      7. log1p-expm1-u 0.4%

         \[\leadsto -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)\right)} \]

      8. log1p-udef 0.6%

         \[\leadsto -\color{blue}{\log \left(1 + \mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)\right)} \]

      9. neg-log 0.6%

         \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)}\right)} \]

      10. add-sqr-sqrt 0.3%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \color{blue}{\left(\sqrt{-\sin re} \cdot \sqrt{-\sin re}\right)}\right)}\right) \]

      11. sqrt-unprod 24.6%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \color{blue}{\sqrt{\left(-\sin re\right) \cdot \left(-\sin re\right)}}\right)}\right) \]

      12. sqr-neg 24.6%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \sqrt{\color{blue}{\sin re \cdot \sin re}}\right)}\right) \]

      13. sqrt-unprod 24.3%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \color{blue}{\left(\sqrt{\sin re} \cdot \sqrt{\sin re}\right)}\right)}\right) \]

      14. add-sqr-sqrt 59.4%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \color{blue}{\sin re}\right)}\right) \]

   6. Applied egg-rr 59.4%

      \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \sin re\right)}\right)} \]

   7. Taylor expanded in re around 0 20.5%

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

      1. associate-+r+ 20.5%

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

      2. mul-1-neg 20.5%

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

      3. unsub-neg 20.5%

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

      4. +-commutative 20.5%

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

      5. mul-1-neg 20.5%

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

      6. unsub-neg 20.5%

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

      7. *-commutative 20.5%

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

      8. distribute-rgt-out 20.5%

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

      9. unpow2 20.5%

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

      10. metadata-eval 20.5%

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

      11. unpow2 20.5%

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

   9. Simplified 20.5%

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

   10. Taylor expanded in re around inf 37.4%

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

       1. *-commutative 37.4%

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

       2. unpow2 37.4%

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

       3. unpow2 37.4%

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

   12. Simplified 37.4%

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

   if -36000 < im < 2.00000000000000001e41

   1. Initial program 34.1%

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

   2. Taylor expanded in im around 0 92.1%

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

      1. mul-1-neg 92.1%

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

      2. *-commutative 92.1%

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

      3. distribute-rgt-neg-in 92.1%

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

   4. Simplified 92.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.75 \cdot 10^{+107}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -36000:\\ \;\;\;\;\log \left(0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+41}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+101}:\\ \;\;\;\;\log \left(0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 6: 60.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{if}\;im \leq -36000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 37000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+189}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (- (* 0.16666666666666666 (pow re 3.0)) re))))
   (if (<= im -36000.0)
     t_0
     (if (<= im 37000000.0)
       (* im (- (sin re)))
       (if (<= im 1.9e+189) t_0 (* im (- re)))))))

C

double code(double re, double im) {
	double t_0 = im * ((0.16666666666666666 * pow(re, 3.0)) - re);
	double tmp;
	if (im <= -36000.0) {
		tmp = t_0;
	} else if (im <= 37000000.0) {
		tmp = im * -sin(re);
	} else if (im <= 1.9e+189) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = im * ((0.16666666666666666d0 * (re ** 3.0d0)) - re)
    if (im <= (-36000.0d0)) then
        tmp = t_0
    else if (im <= 37000000.0d0) then
        tmp = im * -sin(re)
    else if (im <= 1.9d+189) then
        tmp = t_0
    else
        tmp = im * -re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * ((0.16666666666666666 * Math.pow(re, 3.0)) - re);
	double tmp;
	if (im <= -36000.0) {
		tmp = t_0;
	} else if (im <= 37000000.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 1.9e+189) {
		tmp = t_0;
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * ((0.16666666666666666 * math.pow(re, 3.0)) - re)
	tmp = 0
	if im <= -36000.0:
		tmp = t_0
	elif im <= 37000000.0:
		tmp = im * -math.sin(re)
	elif im <= 1.9e+189:
		tmp = t_0
	else:
		tmp = im * -re
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - re))
	tmp = 0.0
	if (im <= -36000.0)
		tmp = t_0;
	elseif (im <= 37000000.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 1.9e+189)
		tmp = t_0;
	else
		tmp = Float64(im * Float64(-re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * ((0.16666666666666666 * (re ^ 3.0)) - re);
	tmp = 0.0;
	if (im <= -36000.0)
		tmp = t_0;
	elseif (im <= 37000000.0)
		tmp = im * -sin(re);
	elseif (im <= 1.9e+189)
		tmp = t_0;
	else
		tmp = im * -re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -36000.0], t$95$0, If[LessEqual[im, 37000000.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.9e+189], t$95$0, N[(im * (-re)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -36000 or 3.7e7 < im < 1.8999999999999999e189

   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 3.8%

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

      1. mul-1-neg 3.8%

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

      2. *-commutative 3.8%

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

      3. distribute-rgt-neg-in 3.8%

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

   4. Simplified 3.8%

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

   5. Taylor expanded in re around 0 14.3%

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

      1. +-commutative 14.3%

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

      2. mul-1-neg 14.3%

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

      3. unsub-neg 14.3%

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

      4. associate-*r* 14.3%

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

      5. distribute-rgt-out-- 22.4%

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

   7. Simplified 22.4%

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

   if -36000 < im < 3.7e7

   1. Initial program 29.7%

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

   2. Taylor expanded in im around 0 98.1%

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

      1. mul-1-neg 98.1%

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

      2. *-commutative 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

   4. Simplified 98.1%

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

   if 1.8999999999999999e189 < 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 5.5%

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

      1. mul-1-neg 5.5%

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

      2. *-commutative 5.5%

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

      3. distribute-rgt-neg-in 5.5%

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

   4. Simplified 5.5%

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

   5. Taylor expanded in re around 0 38.0%

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

      1. mul-1-neg 38.0%

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

      2. distribute-rgt-neg-in 38.0%

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

   7. Simplified 38.0%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -36000:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{elif}\;im \leq 37000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+189}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]

Alternative 7: 77.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -2.05 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -36000:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (- (* (pow im 3.0) -0.16666666666666666) im))))
   (if (<= im -2.05e+102)
     t_0
     (if (<= im -36000.0)
       (* im (- (* 0.16666666666666666 (pow re 3.0)) re))
       (if (<= im 4.2e-7) (* im (- (sin re))) t_0)))))

C

double code(double re, double im) {
	double t_0 = re * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -2.05e+102) {
		tmp = t_0;
	} else if (im <= -36000.0) {
		tmp = im * ((0.16666666666666666 * pow(re, 3.0)) - re);
	} else if (im <= 4.2e-7) {
		tmp = im * -sin(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    if (im <= (-2.05d+102)) then
        tmp = t_0
    else if (im <= (-36000.0d0)) then
        tmp = im * ((0.16666666666666666d0 * (re ** 3.0d0)) - re)
    else if (im <= 4.2d-7) then
        tmp = im * -sin(re)
    else
        tmp = t_0
    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);
	double tmp;
	if (im <= -2.05e+102) {
		tmp = t_0;
	} else if (im <= -36000.0) {
		tmp = im * ((0.16666666666666666 * Math.pow(re, 3.0)) - re);
	} else if (im <= 4.2e-7) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	tmp = 0
	if im <= -2.05e+102:
		tmp = t_0
	elif im <= -36000.0:
		tmp = im * ((0.16666666666666666 * math.pow(re, 3.0)) - re)
	elif im <= 4.2e-7:
		tmp = im * -math.sin(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	tmp = 0.0
	if (im <= -2.05e+102)
		tmp = t_0;
	elseif (im <= -36000.0)
		tmp = Float64(im * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - re));
	elseif (im <= 4.2e-7)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (((im ^ 3.0) * -0.16666666666666666) - im);
	tmp = 0.0;
	if (im <= -2.05e+102)
		tmp = t_0;
	elseif (im <= -36000.0)
		tmp = im * ((0.16666666666666666 * (re ^ 3.0)) - re);
	elseif (im <= 4.2e-7)
		tmp = im * -sin(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.05e+102], t$95$0, If[LessEqual[im, -36000.0], N[(im * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e-7], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.05e102 or 4.2e-7 < im

   1. Initial program 99.7%

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

   2. Taylor expanded in im around 0 80.3%

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

      1. mul-1-neg 80.3%

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

      2. unsub-neg 80.3%

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

      3. *-commutative 80.3%

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

      4. associate-*l* 80.3%

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

      5. distribute-lft-out-- 80.3%

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

   4. Simplified 80.3%

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

   5. Taylor expanded in re around 0 65.9%

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

   if -2.05e102 < im < -36000

   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 3.1%

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

      1. mul-1-neg 3.1%

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

      2. *-commutative 3.1%

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

      3. distribute-rgt-neg-in 3.1%

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

   4. Simplified 3.1%

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

   5. Taylor expanded in re around 0 20.3%

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

      1. +-commutative 20.3%

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

      2. mul-1-neg 20.3%

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

      3. unsub-neg 20.3%

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

      4. associate-*r* 20.3%

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

      5. distribute-rgt-out-- 23.4%

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

   7. Simplified 23.4%

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

   if -36000 < im < 4.2e-7

   1. Initial program 28.7%

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

   2. Taylor expanded in im around 0 99.0%

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

      1. mul-1-neg 99.0%

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

      2. *-commutative 99.0%

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

      3. distribute-rgt-neg-in 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.05 \cdot 10^{+102}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -36000:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 8: 77.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -1.22 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -36000:\\ \;\;\;\;\log \left(0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (- (* (pow im 3.0) -0.16666666666666666) im))))
   (if (<= im -1.22e+77)
     t_0
     (if (<= im -36000.0)
       (log (* 0.5 (* (* re re) (* im im))))
       (if (<= im 4.2e-7) (* im (- (sin re))) t_0)))))

C

double code(double re, double im) {
	double t_0 = re * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -1.22e+77) {
		tmp = t_0;
	} else if (im <= -36000.0) {
		tmp = log((0.5 * ((re * re) * (im * im))));
	} else if (im <= 4.2e-7) {
		tmp = im * -sin(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    if (im <= (-1.22d+77)) then
        tmp = t_0
    else if (im <= (-36000.0d0)) then
        tmp = log((0.5d0 * ((re * re) * (im * im))))
    else if (im <= 4.2d-7) then
        tmp = im * -sin(re)
    else
        tmp = t_0
    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);
	double tmp;
	if (im <= -1.22e+77) {
		tmp = t_0;
	} else if (im <= -36000.0) {
		tmp = Math.log((0.5 * ((re * re) * (im * im))));
	} else if (im <= 4.2e-7) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	tmp = 0
	if im <= -1.22e+77:
		tmp = t_0
	elif im <= -36000.0:
		tmp = math.log((0.5 * ((re * re) * (im * im))))
	elif im <= 4.2e-7:
		tmp = im * -math.sin(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	tmp = 0.0
	if (im <= -1.22e+77)
		tmp = t_0;
	elseif (im <= -36000.0)
		tmp = log(Float64(0.5 * Float64(Float64(re * re) * Float64(im * im))));
	elseif (im <= 4.2e-7)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (((im ^ 3.0) * -0.16666666666666666) - im);
	tmp = 0.0;
	if (im <= -1.22e+77)
		tmp = t_0;
	elseif (im <= -36000.0)
		tmp = log((0.5 * ((re * re) * (im * im))));
	elseif (im <= 4.2e-7)
		tmp = im * -sin(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.22e+77], t$95$0, If[LessEqual[im, -36000.0], N[Log[N[(0.5 * N[(N[(re * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 4.2e-7], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.22000000000000012e77 or 4.2e-7 < im

   1. Initial program 99.7%

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

   2. Taylor expanded in im around 0 75.8%

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

      1. mul-1-neg 75.8%

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

      2. unsub-neg 75.8%

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

      3. *-commutative 75.8%

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

      4. associate-*l* 75.8%

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

      5. distribute-lft-out-- 75.8%

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

   4. Simplified 75.8%

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

   5. Taylor expanded in re around 0 62.9%

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

   if -1.22000000000000012e77 < im < -36000

   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 3.0%

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

      1. mul-1-neg 3.0%

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

      2. *-commutative 3.0%

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

      3. distribute-rgt-neg-in 3.0%

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

   4. Simplified 3.0%

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

      1. distribute-rgt-neg-out 3.0%

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

      2. add-sqr-sqrt 1.3%

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

      3. sqrt-unprod 1.8%

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

      4. sqr-neg 1.8%

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

      5. sqrt-unprod 0.4%

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

      6. add-sqr-sqrt 0.7%

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

      7. log1p-expm1-u 0.4%

         \[\leadsto -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)\right)} \]

      8. log1p-udef 0.6%

         \[\leadsto -\color{blue}{\log \left(1 + \mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)\right)} \]

      9. neg-log 0.6%

         \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)}\right)} \]

      10. add-sqr-sqrt 0.3%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \color{blue}{\left(\sqrt{-\sin re} \cdot \sqrt{-\sin re}\right)}\right)}\right) \]

      11. sqrt-unprod 28.7%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \color{blue}{\sqrt{\left(-\sin re\right) \cdot \left(-\sin re\right)}}\right)}\right) \]

      12. sqr-neg 28.7%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \sqrt{\color{blue}{\sin re \cdot \sin re}}\right)}\right) \]

      13. sqrt-unprod 28.3%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \color{blue}{\left(\sqrt{\sin re} \cdot \sqrt{\sin re}\right)}\right)}\right) \]

      14. add-sqr-sqrt 60.7%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \color{blue}{\sin re}\right)}\right) \]

   6. Applied egg-rr 60.7%

      \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \sin re\right)}\right)} \]

   7. Taylor expanded in re around 0 21.0%

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

      1. associate-+r+ 21.0%

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

      2. mul-1-neg 21.0%

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

      3. unsub-neg 21.0%

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

      4. +-commutative 21.0%

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

      5. mul-1-neg 21.0%

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

      6. unsub-neg 21.0%

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

      7. *-commutative 21.0%

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

      8. distribute-rgt-out 21.0%

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

      9. unpow2 21.0%

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

      10. metadata-eval 21.0%

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

      11. unpow2 21.0%

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

   9. Simplified 21.0%

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

   10. Taylor expanded in re around inf 40.5%

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

       1. *-commutative 40.5%

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

       2. unpow2 40.5%

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

       3. unpow2 40.5%

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

   12. Simplified 40.5%

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

   if -36000 < im < 4.2e-7

   1. Initial program 28.7%

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

   2. Taylor expanded in im around 0 99.0%

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

      1. mul-1-neg 99.0%

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

      2. *-commutative 99.0%

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

      3. distribute-rgt-neg-in 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.22 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -36000:\\ \;\;\;\;\log \left(0.5 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 9: 60.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{if}\;im \leq -36000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 15800000000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+190}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.16666666666666666 (* im (pow re 3.0)))))
   (if (<= im -36000.0)
     t_0
     (if (<= im 15800000000000.0)
       (* im (- (sin re)))
       (if (<= im 2.1e+190) t_0 (* im (- re)))))))

C

double code(double re, double im) {
	double t_0 = 0.16666666666666666 * (im * pow(re, 3.0));
	double tmp;
	if (im <= -36000.0) {
		tmp = t_0;
	} else if (im <= 15800000000000.0) {
		tmp = im * -sin(re);
	} else if (im <= 2.1e+190) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 * (im * (re ** 3.0d0))
    if (im <= (-36000.0d0)) then
        tmp = t_0
    else if (im <= 15800000000000.0d0) then
        tmp = im * -sin(re)
    else if (im <= 2.1d+190) then
        tmp = t_0
    else
        tmp = im * -re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.16666666666666666 * (im * Math.pow(re, 3.0));
	double tmp;
	if (im <= -36000.0) {
		tmp = t_0;
	} else if (im <= 15800000000000.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 2.1e+190) {
		tmp = t_0;
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.16666666666666666 * (im * math.pow(re, 3.0))
	tmp = 0
	if im <= -36000.0:
		tmp = t_0
	elif im <= 15800000000000.0:
		tmp = im * -math.sin(re)
	elif im <= 2.1e+190:
		tmp = t_0
	else:
		tmp = im * -re
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.16666666666666666 * Float64(im * (re ^ 3.0)))
	tmp = 0.0
	if (im <= -36000.0)
		tmp = t_0;
	elseif (im <= 15800000000000.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 2.1e+190)
		tmp = t_0;
	else
		tmp = Float64(im * Float64(-re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.16666666666666666 * (im * (re ^ 3.0));
	tmp = 0.0;
	if (im <= -36000.0)
		tmp = t_0;
	elseif (im <= 15800000000000.0)
		tmp = im * -sin(re);
	elseif (im <= 2.1e+190)
		tmp = t_0;
	else
		tmp = im * -re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(im * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -36000.0], t$95$0, If[LessEqual[im, 15800000000000.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 2.1e+190], t$95$0, N[(im * (-re)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -36000 or 1.58e13 < im < 2.1000000000000001e190

   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 3.8%

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

      1. mul-1-neg 3.8%

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

      2. *-commutative 3.8%

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

      3. distribute-rgt-neg-in 3.8%

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

   4. Simplified 3.8%

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

   5. Taylor expanded in re around 0 14.6%

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

   6. Taylor expanded in re around inf 21.8%

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

   if -36000 < im < 1.58e13

   1. Initial program 30.8%

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

   2. Taylor expanded in im around 0 96.5%

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

      1. mul-1-neg 96.5%

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

      2. *-commutative 96.5%

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

      3. distribute-rgt-neg-in 96.5%

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

   4. Simplified 96.5%

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

   if 2.1000000000000001e190 < 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 5.5%

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

      1. mul-1-neg 5.5%

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

      2. *-commutative 5.5%

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

      3. distribute-rgt-neg-in 5.5%

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

   4. Simplified 5.5%

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

   5. Taylor expanded in re around 0 38.0%

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

      1. mul-1-neg 38.0%

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

      2. distribute-rgt-neg-in 38.0%

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

   7. Simplified 38.0%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -36000:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{elif}\;im \leq 15800000000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+190}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]

Alternative 10: 54.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2 \cdot 10^{+26}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.2e+26) (* im (- (sin re))) (* im (- re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.2e+26) {
		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 <= 6.2d+26) 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 <= 6.2e+26) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.2e+26:
		tmp = im * -math.sin(re)
	else:
		tmp = im * -re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.2e+26)
		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 <= 6.2e+26)
		tmp = im * -sin(re);
	else
		tmp = im * -re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.2e+26], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(im * (-re)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.1999999999999999e26

   1. Initial program 56.8%

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

   2. Taylor expanded in im around 0 61.6%

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

      1. mul-1-neg 61.6%

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

      2. *-commutative 61.6%

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

      3. distribute-rgt-neg-in 61.6%

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

   4. Simplified 61.6%

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

   if 6.1999999999999999e26 < 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 4.4%

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

      1. mul-1-neg 4.4%

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

      2. *-commutative 4.4%

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

      3. distribute-rgt-neg-in 4.4%

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

   4. Simplified 4.4%

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

   5. Taylor expanded in re around 0 24.7%

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

      1. mul-1-neg 24.7%

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

      2. distribute-rgt-neg-in 24.7%

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

   7. Simplified 24.7%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2 \cdot 10^{+26}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]

Alternative 11: 32.7% 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 67.3%

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

2. Taylor expanded in im around 0 47.8%

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

   1. mul-1-neg 47.8%

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

   2. *-commutative 47.8%

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

   3. distribute-rgt-neg-in 47.8%

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

4. Simplified 47.8%

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

5. Taylor expanded in re around 0 34.7%

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

   1. mul-1-neg 34.7%

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

   2. distribute-rgt-neg-in 34.7%

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

7. Simplified 34.7%

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

8. Final simplification 34.7%

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

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 2023257 
(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))))