math.cos on complex, imaginary part

Percentage Accurate: 64.5% → 99.8%

Time: 8.9s

Alternatives: 18

Speedup: 2.8×

• 47.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 64.5% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.2 \lor \neg \left(t_0 \leq 10^{-7}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right) - im \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.2) (not (<= t_0 1e-7)))
     (* t_0 (* 0.5 (sin re)))
     (- (* (pow im 3.0) (* (sin re) -0.16666666666666666)) (* im (sin re))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 1e-7)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = (pow(im, 3.0) * (sin(re) * -0.16666666666666666)) - (im * sin(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.2d0)) .or. (.not. (t_0 <= 1d-7))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = ((im ** 3.0d0) * (sin(re) * (-0.16666666666666666d0))) - (im * sin(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 1e-7)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = (Math.pow(im, 3.0) * (Math.sin(re) * -0.16666666666666666)) - (im * Math.sin(re));
	}
	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 <= 1e-7):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = (math.pow(im, 3.0) * (math.sin(re) * -0.16666666666666666)) - (im * math.sin(re))
	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 <= 1e-7))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(Float64((im ^ 3.0) * Float64(sin(re) * -0.16666666666666666)) - Float64(im * sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.2) || ~((t_0 <= 1e-7)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = ((im ^ 3.0) * (sin(re) * -0.16666666666666666)) - (im * sin(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.2], N[Not[LessEqual[t$95$0, 1e-7]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[im, 3.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(im * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.20000000000000001 or 9.9999999999999995e-8 < (-.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)) < 9.9999999999999995e-8

   1. Initial program 30.9%

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

   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. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

      6. *-commutative 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.2 \lor \neg \left(e^{-im} - e^{im} \leq 10^{-7}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right) - im \cdot \sin re\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.2 \lor \neg \left(t_0 \leq 10^{-7}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.2) (not (<= t_0 1e-7)))
     (* 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 <= 1e-7)) {
		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 <= 1d-7))) 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 <= 1e-7)) {
		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 <= 1e-7):
		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 <= 1e-7))
		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 <= 1e-7)))
		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, 1e-7]], $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 9.9999999999999995e-8 < (-.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)) < 9.9999999999999995e-8

   1. Initial program 30.9%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.2 \lor \neg \left(e^{-im} - e^{im} \leq 10^{-7}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 95.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := {im}^{3} \cdot -0.16666666666666666\\ t_2 := \sin re \cdot t_1\\ \mathbf{if}\;im \leq -2.35 \cdot 10^{+103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -0.024:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.043:\\ \;\;\;\;\sin re \cdot \left(t_1 - im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im)) (exp im)) (* 0.5 re)))
        (t_1 (* (pow im 3.0) -0.16666666666666666))
        (t_2 (* (sin re) t_1)))
   (if (<= im -2.35e+103)
     t_2
     (if (<= im -0.024)
       t_0
       (if (<= im 0.043)
         (* (sin re) (- t_1 im))
         (if (<= im 2e+102) t_0 t_2))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) - exp(im)) * (0.5 * re);
	double t_1 = pow(im, 3.0) * -0.16666666666666666;
	double t_2 = sin(re) * t_1;
	double tmp;
	if (im <= -2.35e+103) {
		tmp = t_2;
	} else if (im <= -0.024) {
		tmp = t_0;
	} else if (im <= 0.043) {
		tmp = sin(re) * (t_1 - im);
	} else if (im <= 2e+102) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (exp(-im) - exp(im)) * (0.5d0 * re)
    t_1 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_2 = sin(re) * t_1
    if (im <= (-2.35d+103)) then
        tmp = t_2
    else if (im <= (-0.024d0)) then
        tmp = t_0
    else if (im <= 0.043d0) then
        tmp = sin(re) * (t_1 - im)
    else if (im <= 2d+102) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.exp(-im) - Math.exp(im)) * (0.5 * re);
	double t_1 = Math.pow(im, 3.0) * -0.16666666666666666;
	double t_2 = Math.sin(re) * t_1;
	double tmp;
	if (im <= -2.35e+103) {
		tmp = t_2;
	} else if (im <= -0.024) {
		tmp = t_0;
	} else if (im <= 0.043) {
		tmp = Math.sin(re) * (t_1 - im);
	} else if (im <= 2e+102) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) - math.exp(im)) * (0.5 * re)
	t_1 = math.pow(im, 3.0) * -0.16666666666666666
	t_2 = math.sin(re) * t_1
	tmp = 0
	if im <= -2.35e+103:
		tmp = t_2
	elif im <= -0.024:
		tmp = t_0
	elif im <= 0.043:
		tmp = math.sin(re) * (t_1 - im)
	elif im <= 2e+102:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 * re))
	t_1 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_2 = Float64(sin(re) * t_1)
	tmp = 0.0
	if (im <= -2.35e+103)
		tmp = t_2;
	elseif (im <= -0.024)
		tmp = t_0;
	elseif (im <= 0.043)
		tmp = Float64(sin(re) * Float64(t_1 - im));
	elseif (im <= 2e+102)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) - exp(im)) * (0.5 * re);
	t_1 = (im ^ 3.0) * -0.16666666666666666;
	t_2 = sin(re) * t_1;
	tmp = 0.0;
	if (im <= -2.35e+103)
		tmp = t_2;
	elseif (im <= -0.024)
		tmp = t_0;
	elseif (im <= 0.043)
		tmp = sin(re) * (t_1 - im);
	elseif (im <= 2e+102)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[re], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[im, -2.35e+103], t$95$2, If[LessEqual[im, -0.024], t$95$0, If[LessEqual[im, 0.043], N[(N[Sin[re], $MachinePrecision] * N[(t$95$1 - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2e+102], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.35000000000000016e103 or 1.99999999999999995e102 < 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%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -2.35000000000000016e103 < im < -0.024 or 0.042999999999999997 < im < 1.99999999999999995e102

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 79.9%

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

   if -0.024 < im < 0.042999999999999997

   1. Initial program 30.9%

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

   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 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.35 \cdot 10^{+103}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -0.024:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 0.043:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 4: 85.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{if}\;im \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -850000000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot 13.5\right)\right)\\ \mathbf{elif}\;im \leq 2.4:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) (* (pow im 3.0) -0.16666666666666666))))
   (if (<= im -5.6e+102)
     t_0
     (if (<= im -850000000000.0)
       (log1p (expm1 (* re 13.5)))
       (if (<= im 2.4) (* im (- (sin re))) t_0)))))

C

double code(double re, double im) {
	double t_0 = sin(re) * (pow(im, 3.0) * -0.16666666666666666);
	double tmp;
	if (im <= -5.6e+102) {
		tmp = t_0;
	} else if (im <= -850000000000.0) {
		tmp = log1p(expm1((re * 13.5)));
	} else if (im <= 2.4) {
		tmp = im * -sin(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.sin(re) * (Math.pow(im, 3.0) * -0.16666666666666666);
	double tmp;
	if (im <= -5.6e+102) {
		tmp = t_0;
	} else if (im <= -850000000000.0) {
		tmp = Math.log1p(Math.expm1((re * 13.5)));
	} else if (im <= 2.4) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sin(re) * (math.pow(im, 3.0) * -0.16666666666666666)
	tmp = 0
	if im <= -5.6e+102:
		tmp = t_0
	elif im <= -850000000000.0:
		tmp = math.log1p(math.expm1((re * 13.5)))
	elif im <= 2.4:
		tmp = im * -math.sin(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(sin(re) * Float64((im ^ 3.0) * -0.16666666666666666))
	tmp = 0.0
	if (im <= -5.6e+102)
		tmp = t_0;
	elseif (im <= -850000000000.0)
		tmp = log1p(expm1(Float64(re * 13.5)));
	elseif (im <= 2.4)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5.6e+102], t$95$0, If[LessEqual[im, -850000000000.0], N[Log[1 + N[(Exp[N[(re * 13.5), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 2.4], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.60000000000000037e102 or 2.39999999999999991 < 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 88.6%

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

      1. mul-1-neg 88.6%

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

      2. unsub-neg 88.6%

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

      3. *-commutative 88.6%

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

      4. associate-*l* 88.6%

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

      5. distribute-lft-out-- 88.6%

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

   4. Simplified 88.6%

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

   5. Taylor expanded in im around inf 88.6%

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

      1. *-commutative 88.6%

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

      2. associate-*l* 88.6%

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

      3. *-commutative 88.6%

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

   7. Simplified 88.6%

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

   if -5.60000000000000037e102 < im < -8.5e11

   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 73.9%

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

   4. Applied egg-rr 2.7%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{27} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 35.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(0.5 \cdot re\right) \cdot 27\right)\right)} \]

      2. *-commutative 35.5%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(re \cdot 0.5\right)} \cdot 27\right)\right) \]

      3. associate-*l* 35.5%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{re \cdot \left(0.5 \cdot 27\right)}\right)\right) \]

      4. metadata-eval 35.5%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \color{blue}{13.5}\right)\right) \]

   6. Applied egg-rr 35.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot 13.5\right)\right)} \]

   if -8.5e11 < im < 2.39999999999999991

   1. Initial program 32.9%

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

   2. Taylor expanded in im around 0 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

   4. Simplified 97.1%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -850000000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot 13.5\right)\right)\\ \mathbf{elif}\;im \leq 2.4:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 5: 76.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+135}:\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -7.8 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -1.5\right)\right)\\ \mathbf{elif}\;im \leq -950000000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot 13.5\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666\right) - im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -6e+135)
   (* (pow im 3.0) (* re -0.16666666666666666))
   (if (<= im -7.8e+105)
     (log1p (expm1 (* re -1.5)))
     (if (<= im -950000000000.0)
       (log1p (expm1 (* re 13.5)))
       (if (<= im 4.2e-5)
         (* im (- (sin re)))
         (- (* re (* (pow im 3.0) -0.16666666666666666)) (* im re)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -6e+135) {
		tmp = pow(im, 3.0) * (re * -0.16666666666666666);
	} else if (im <= -7.8e+105) {
		tmp = log1p(expm1((re * -1.5)));
	} else if (im <= -950000000000.0) {
		tmp = log1p(expm1((re * 13.5)));
	} else if (im <= 4.2e-5) {
		tmp = im * -sin(re);
	} else {
		tmp = (re * (pow(im, 3.0) * -0.16666666666666666)) - (im * re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -6e+135) {
		tmp = Math.pow(im, 3.0) * (re * -0.16666666666666666);
	} else if (im <= -7.8e+105) {
		tmp = Math.log1p(Math.expm1((re * -1.5)));
	} else if (im <= -950000000000.0) {
		tmp = Math.log1p(Math.expm1((re * 13.5)));
	} else if (im <= 4.2e-5) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = (re * (Math.pow(im, 3.0) * -0.16666666666666666)) - (im * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -6e+135:
		tmp = math.pow(im, 3.0) * (re * -0.16666666666666666)
	elif im <= -7.8e+105:
		tmp = math.log1p(math.expm1((re * -1.5)))
	elif im <= -950000000000.0:
		tmp = math.log1p(math.expm1((re * 13.5)))
	elif im <= 4.2e-5:
		tmp = im * -math.sin(re)
	else:
		tmp = (re * (math.pow(im, 3.0) * -0.16666666666666666)) - (im * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -6e+135)
		tmp = Float64((im ^ 3.0) * Float64(re * -0.16666666666666666));
	elseif (im <= -7.8e+105)
		tmp = log1p(expm1(Float64(re * -1.5)));
	elseif (im <= -950000000000.0)
		tmp = log1p(expm1(Float64(re * 13.5)));
	elseif (im <= 4.2e-5)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = Float64(Float64(re * Float64((im ^ 3.0) * -0.16666666666666666)) - Float64(im * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -6e+135], N[(N[Power[im, 3.0], $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -7.8e+105], N[Log[1 + N[(Exp[N[(re * -1.5), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, -950000000000.0], N[Log[1 + N[(Exp[N[(re * 13.5), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 4.2e-5], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(N[(re * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(im * re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -6.0000000000000001e135

   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 80.6%

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

   3. Taylor expanded in im around 0 80.6%

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

   4. Taylor expanded in im around inf 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-*r* 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*r* 80.6%

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

   6. Simplified 80.6%

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

   if -6.0000000000000001e135 < im < -7.79999999999999957e105

   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 27.3%

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

   4. Applied egg-rr 4.7%

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

      1. log1p-expm1-u 72.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(0.5 \cdot re\right) \cdot -3\right)\right)} \]

      2. *-commutative 72.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(re \cdot 0.5\right)} \cdot -3\right)\right) \]

      3. associate-*l* 72.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{re \cdot \left(0.5 \cdot -3\right)}\right)\right) \]

      4. metadata-eval 72.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \color{blue}{-1.5}\right)\right) \]

   6. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -1.5\right)\right)} \]

   if -7.79999999999999957e105 < im < -9.5e11

   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 75.0%

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

   4. Applied egg-rr 2.7%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{27} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 34.1%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(0.5 \cdot re\right) \cdot 27\right)\right)} \]

      2. *-commutative 34.1%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(re \cdot 0.5\right)} \cdot 27\right)\right) \]

      3. associate-*l* 34.1%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{re \cdot \left(0.5 \cdot 27\right)}\right)\right) \]

      4. metadata-eval 34.1%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \color{blue}{13.5}\right)\right) \]

   6. Applied egg-rr 34.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot 13.5\right)\right)} \]

   if -9.5e11 < im < 4.19999999999999977e-5

   1. Initial program 32.1%

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

   2. Taylor expanded in im around 0 97.8%

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

      1. mul-1-neg 97.8%

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

      2. *-commutative 97.8%

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

      3. distribute-rgt-neg-in 97.8%

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

   4. Simplified 97.8%

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

   if 4.19999999999999977e-5 < im

   1. Initial program 99.6%

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

   2. Taylor expanded in re around 0 71.8%

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

   3. Taylor expanded in im around 0 55.6%

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

      1. +-commutative 55.6%

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

      2. mul-1-neg 55.6%

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

      3. unsub-neg 55.6%

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

      4. *-commutative 55.6%

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

      5. associate-*l* 55.6%

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

   5. Applied egg-rr 55.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+135}:\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -7.8 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -1.5\right)\right)\\ \mathbf{elif}\;im \leq -950000000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot 13.5\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666\right) - im \cdot re\\ \end{array} \]

Alternative 6: 84.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im)))

C

double code(double re, double im) {
	return sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
end function

Java

public static double code(double re, double im) {
	return Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
}

Python

def code(re, im):
	return math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)

Julia

function code(re, im)
	return Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

2. Taylor expanded in im around 0 85.6%

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

   1. mul-1-neg 85.6%

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

   2. unsub-neg 85.6%

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

   3. *-commutative 85.6%

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

   4. associate-*l* 85.6%

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

   5. distribute-lft-out-- 85.6%

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

4. Simplified 85.6%

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

5. Final simplification 85.6%

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

Alternative 7: 76.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+135}:\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -8.1 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -1.5\right)\right)\\ \mathbf{elif}\;im \leq -0.098 \lor \neg \left(im \leq 4.7 \cdot 10^{-5}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666\right) - im \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -6e+135)
   (* (pow im 3.0) (* re -0.16666666666666666))
   (if (<= im -8.1e+105)
     (log1p (expm1 (* re -1.5)))
     (if (or (<= im -0.098) (not (<= im 4.7e-5)))
       (- (* re (* (pow im 3.0) -0.16666666666666666)) (* im re))
       (* im (- (sin re)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -6e+135) {
		tmp = pow(im, 3.0) * (re * -0.16666666666666666);
	} else if (im <= -8.1e+105) {
		tmp = log1p(expm1((re * -1.5)));
	} else if ((im <= -0.098) || !(im <= 4.7e-5)) {
		tmp = (re * (pow(im, 3.0) * -0.16666666666666666)) - (im * re);
	} else {
		tmp = im * -sin(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -6e+135) {
		tmp = Math.pow(im, 3.0) * (re * -0.16666666666666666);
	} else if (im <= -8.1e+105) {
		tmp = Math.log1p(Math.expm1((re * -1.5)));
	} else if ((im <= -0.098) || !(im <= 4.7e-5)) {
		tmp = (re * (Math.pow(im, 3.0) * -0.16666666666666666)) - (im * re);
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -6e+135:
		tmp = math.pow(im, 3.0) * (re * -0.16666666666666666)
	elif im <= -8.1e+105:
		tmp = math.log1p(math.expm1((re * -1.5)))
	elif (im <= -0.098) or not (im <= 4.7e-5):
		tmp = (re * (math.pow(im, 3.0) * -0.16666666666666666)) - (im * re)
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -6e+135)
		tmp = Float64((im ^ 3.0) * Float64(re * -0.16666666666666666));
	elseif (im <= -8.1e+105)
		tmp = log1p(expm1(Float64(re * -1.5)));
	elseif ((im <= -0.098) || !(im <= 4.7e-5))
		tmp = Float64(Float64(re * Float64((im ^ 3.0) * -0.16666666666666666)) - Float64(im * re));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -6e+135], N[(N[Power[im, 3.0], $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -8.1e+105], N[Log[1 + N[(Exp[N[(re * -1.5), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[im, -0.098], N[Not[LessEqual[im, 4.7e-5]], $MachinePrecision]], N[(N[(re * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(im * re), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -6.0000000000000001e135

   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 80.6%

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

   3. Taylor expanded in im around 0 80.6%

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

   4. Taylor expanded in im around inf 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-*r* 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*r* 80.6%

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

   6. Simplified 80.6%

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

   if -6.0000000000000001e135 < im < -8.09999999999999998e105

   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 27.3%

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

   4. Applied egg-rr 4.7%

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

      1. log1p-expm1-u 72.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(0.5 \cdot re\right) \cdot -3\right)\right)} \]

      2. *-commutative 72.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(re \cdot 0.5\right)} \cdot -3\right)\right) \]

      3. associate-*l* 72.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{re \cdot \left(0.5 \cdot -3\right)}\right)\right) \]

      4. metadata-eval 72.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \color{blue}{-1.5}\right)\right) \]

   6. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -1.5\right)\right)} \]

   if -8.09999999999999998e105 < im < -0.098000000000000004 or 4.69999999999999972e-5 < im

   1. Initial program 99.7%

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

   2. Taylor expanded in re around 0 73.6%

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

   3. Taylor expanded in im around 0 47.5%

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

      1. +-commutative 47.5%

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

      2. mul-1-neg 47.5%

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

      3. unsub-neg 47.5%

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

      4. *-commutative 47.5%

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

      5. associate-*l* 47.5%

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

   5. Applied egg-rr 47.5%

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

   if -0.098000000000000004 < im < 4.69999999999999972e-5

   1. Initial program 30.5%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+135}:\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -8.1 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot -1.5\right)\right)\\ \mathbf{elif}\;im \leq -0.098 \lor \neg \left(im \leq 4.7 \cdot 10^{-5}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666\right) - im \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 8: 76.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+135}:\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -8.1 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{elif}\;im \leq -0.0088 \lor \neg \left(im \leq 4.7 \cdot 10^{-5}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666\right) - im \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -6e+135)
   (* (pow im 3.0) (* re -0.16666666666666666))
   (if (<= im -8.1e+105)
     (* im (- (* 0.16666666666666666 (pow re 3.0)) re))
     (if (or (<= im -0.0088) (not (<= im 4.7e-5)))
       (- (* re (* (pow im 3.0) -0.16666666666666666)) (* im re))
       (* im (- (sin re)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -6e+135) {
		tmp = pow(im, 3.0) * (re * -0.16666666666666666);
	} else if (im <= -8.1e+105) {
		tmp = im * ((0.16666666666666666 * pow(re, 3.0)) - re);
	} else if ((im <= -0.0088) || !(im <= 4.7e-5)) {
		tmp = (re * (pow(im, 3.0) * -0.16666666666666666)) - (im * re);
	} 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 <= (-6d+135)) then
        tmp = (im ** 3.0d0) * (re * (-0.16666666666666666d0))
    else if (im <= (-8.1d+105)) then
        tmp = im * ((0.16666666666666666d0 * (re ** 3.0d0)) - re)
    else if ((im <= (-0.0088d0)) .or. (.not. (im <= 4.7d-5))) then
        tmp = (re * ((im ** 3.0d0) * (-0.16666666666666666d0))) - (im * re)
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -6e+135) {
		tmp = Math.pow(im, 3.0) * (re * -0.16666666666666666);
	} else if (im <= -8.1e+105) {
		tmp = im * ((0.16666666666666666 * Math.pow(re, 3.0)) - re);
	} else if ((im <= -0.0088) || !(im <= 4.7e-5)) {
		tmp = (re * (Math.pow(im, 3.0) * -0.16666666666666666)) - (im * re);
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -6e+135:
		tmp = math.pow(im, 3.0) * (re * -0.16666666666666666)
	elif im <= -8.1e+105:
		tmp = im * ((0.16666666666666666 * math.pow(re, 3.0)) - re)
	elif (im <= -0.0088) or not (im <= 4.7e-5):
		tmp = (re * (math.pow(im, 3.0) * -0.16666666666666666)) - (im * re)
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -6e+135)
		tmp = Float64((im ^ 3.0) * Float64(re * -0.16666666666666666));
	elseif (im <= -8.1e+105)
		tmp = Float64(im * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - re));
	elseif ((im <= -0.0088) || !(im <= 4.7e-5))
		tmp = Float64(Float64(re * Float64((im ^ 3.0) * -0.16666666666666666)) - Float64(im * re));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -6e+135)
		tmp = (im ^ 3.0) * (re * -0.16666666666666666);
	elseif (im <= -8.1e+105)
		tmp = im * ((0.16666666666666666 * (re ^ 3.0)) - re);
	elseif ((im <= -0.0088) || ~((im <= 4.7e-5)))
		tmp = (re * ((im ^ 3.0) * -0.16666666666666666)) - (im * re);
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -6e+135], N[(N[Power[im, 3.0], $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -8.1e+105], N[(im * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -0.0088], N[Not[LessEqual[im, 4.7e-5]], $MachinePrecision]], N[(N[(re * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(im * re), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -6.0000000000000001e135

   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 80.6%

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

   3. Taylor expanded in im around 0 80.6%

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

   4. Taylor expanded in im around inf 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-*r* 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*r* 80.6%

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

   6. Simplified 80.6%

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

   if -6.0000000000000001e135 < im < -8.09999999999999998e105

   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%

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

      1. mul-1-neg 4.2%

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

      2. *-commutative 4.2%

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

      3. distribute-rgt-neg-in 4.2%

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

   4. Simplified 4.2%

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

   5. Taylor expanded in re around 0 37.1%

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

      1. +-commutative 37.1%

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

      2. mul-1-neg 37.1%

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

      3. unsub-neg 37.1%

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

      4. associate-*r* 37.1%

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

      5. distribute-rgt-out-- 64.4%

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

   7. Simplified 64.4%

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

   if -8.09999999999999998e105 < im < -0.00880000000000000053 or 4.69999999999999972e-5 < im

   1. Initial program 99.7%

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

   2. Taylor expanded in re around 0 73.6%

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

   3. Taylor expanded in im around 0 47.5%

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

      1. +-commutative 47.5%

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

      2. mul-1-neg 47.5%

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

      3. unsub-neg 47.5%

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

      4. *-commutative 47.5%

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

      5. associate-*l* 47.5%

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

   5. Applied egg-rr 47.5%

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

   if -0.00880000000000000053 < im < 4.69999999999999972e-5

   1. Initial program 30.5%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+135}:\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -8.1 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{elif}\;im \leq -0.0088 \lor \neg \left(im \leq 4.7 \cdot 10^{-5}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666\right) - im \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 9: 76.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+135}:\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -8.1 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{elif}\;im \leq -0.0016 \lor \neg \left(im \leq 2.3 \cdot 10^{-5}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -6e+135)
   (* (pow im 3.0) (* re -0.16666666666666666))
   (if (<= im -8.1e+105)
     (* im (- (* 0.16666666666666666 (pow re 3.0)) re))
     (if (or (<= im -0.0016) (not (<= im 2.3e-5)))
       (* re (- (* (pow im 3.0) -0.16666666666666666) im))
       (* im (- (sin re)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -6e+135) {
		tmp = pow(im, 3.0) * (re * -0.16666666666666666);
	} else if (im <= -8.1e+105) {
		tmp = im * ((0.16666666666666666 * pow(re, 3.0)) - re);
	} else if ((im <= -0.0016) || !(im <= 2.3e-5)) {
		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 <= (-6d+135)) then
        tmp = (im ** 3.0d0) * (re * (-0.16666666666666666d0))
    else if (im <= (-8.1d+105)) then
        tmp = im * ((0.16666666666666666d0 * (re ** 3.0d0)) - re)
    else if ((im <= (-0.0016d0)) .or. (.not. (im <= 2.3d-5))) 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 <= -6e+135) {
		tmp = Math.pow(im, 3.0) * (re * -0.16666666666666666);
	} else if (im <= -8.1e+105) {
		tmp = im * ((0.16666666666666666 * Math.pow(re, 3.0)) - re);
	} else if ((im <= -0.0016) || !(im <= 2.3e-5)) {
		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 <= -6e+135:
		tmp = math.pow(im, 3.0) * (re * -0.16666666666666666)
	elif im <= -8.1e+105:
		tmp = im * ((0.16666666666666666 * math.pow(re, 3.0)) - re)
	elif (im <= -0.0016) or not (im <= 2.3e-5):
		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 <= -6e+135)
		tmp = Float64((im ^ 3.0) * Float64(re * -0.16666666666666666));
	elseif (im <= -8.1e+105)
		tmp = Float64(im * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - re));
	elseif ((im <= -0.0016) || !(im <= 2.3e-5))
		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 <= -6e+135)
		tmp = (im ^ 3.0) * (re * -0.16666666666666666);
	elseif (im <= -8.1e+105)
		tmp = im * ((0.16666666666666666 * (re ^ 3.0)) - re);
	elseif ((im <= -0.0016) || ~((im <= 2.3e-5)))
		tmp = re * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -6e+135], N[(N[Power[im, 3.0], $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -8.1e+105], N[(im * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -0.0016], N[Not[LessEqual[im, 2.3e-5]], $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 4 regimes

2. if im < -6.0000000000000001e135

   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 80.6%

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

   3. Taylor expanded in im around 0 80.6%

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

   4. Taylor expanded in im around inf 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-*r* 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*r* 80.6%

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

   6. Simplified 80.6%

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

   if -6.0000000000000001e135 < im < -8.09999999999999998e105

   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%

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

      1. mul-1-neg 4.2%

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

      2. *-commutative 4.2%

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

      3. distribute-rgt-neg-in 4.2%

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

   4. Simplified 4.2%

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

   5. Taylor expanded in re around 0 37.1%

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

      1. +-commutative 37.1%

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

      2. mul-1-neg 37.1%

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

      3. unsub-neg 37.1%

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

      4. associate-*r* 37.1%

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

      5. distribute-rgt-out-- 64.4%

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

   7. Simplified 64.4%

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

   if -8.09999999999999998e105 < im < -0.00160000000000000008 or 2.3e-5 < 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 58.4%

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

      1. mul-1-neg 58.4%

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

      2. unsub-neg 58.4%

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

      3. *-commutative 58.4%

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

      4. associate-*l* 58.4%

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

      5. distribute-lft-out-- 58.4%

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

   4. Simplified 58.4%

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

   5. Taylor expanded in re around 0 47.5%

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

   if -0.00160000000000000008 < im < 2.3e-5

   1. Initial program 30.5%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+135}:\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -8.1 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{elif}\;im \leq -0.0016 \lor \neg \left(im \leq 2.3 \cdot 10^{-5}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 10: 76.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{if}\;im \leq -6 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8.1 \cdot 10^{+105}:\\ \;\;\;\;{re}^{3} \cdot \left(im \cdot 0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -4.2 \cdot 10^{+20} \lor \neg \left(im \leq 700\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) (* re -0.16666666666666666))))
   (if (<= im -6e+135)
     t_0
     (if (<= im -8.1e+105)
       (* (pow re 3.0) (* im 0.16666666666666666))
       (if (or (<= im -4.2e+20) (not (<= im 700.0)))
         t_0
         (* im (- (sin re))))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * (re * -0.16666666666666666);
	double tmp;
	if (im <= -6e+135) {
		tmp = t_0;
	} else if (im <= -8.1e+105) {
		tmp = pow(re, 3.0) * (im * 0.16666666666666666);
	} else if ((im <= -4.2e+20) || !(im <= 700.0)) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (re * (-0.16666666666666666d0))
    if (im <= (-6d+135)) then
        tmp = t_0
    else if (im <= (-8.1d+105)) then
        tmp = (re ** 3.0d0) * (im * 0.16666666666666666d0)
    else if ((im <= (-4.2d+20)) .or. (.not. (im <= 700.0d0))) then
        tmp = t_0
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * (re * -0.16666666666666666);
	double tmp;
	if (im <= -6e+135) {
		tmp = t_0;
	} else if (im <= -8.1e+105) {
		tmp = Math.pow(re, 3.0) * (im * 0.16666666666666666);
	} else if ((im <= -4.2e+20) || !(im <= 700.0)) {
		tmp = t_0;
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * (re * -0.16666666666666666)
	tmp = 0
	if im <= -6e+135:
		tmp = t_0
	elif im <= -8.1e+105:
		tmp = math.pow(re, 3.0) * (im * 0.16666666666666666)
	elif (im <= -4.2e+20) or not (im <= 700.0):
		tmp = t_0
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * Float64(re * -0.16666666666666666))
	tmp = 0.0
	if (im <= -6e+135)
		tmp = t_0;
	elseif (im <= -8.1e+105)
		tmp = Float64((re ^ 3.0) * Float64(im * 0.16666666666666666));
	elseif ((im <= -4.2e+20) || !(im <= 700.0))
		tmp = t_0;
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * (re * -0.16666666666666666);
	tmp = 0.0;
	if (im <= -6e+135)
		tmp = t_0;
	elseif (im <= -8.1e+105)
		tmp = (re ^ 3.0) * (im * 0.16666666666666666);
	elseif ((im <= -4.2e+20) || ~((im <= 700.0)))
		tmp = t_0;
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6e+135], t$95$0, If[LessEqual[im, -8.1e+105], N[(N[Power[re, 3.0], $MachinePrecision] * N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -4.2e+20], N[Not[LessEqual[im, 700.0]], $MachinePrecision]], t$95$0, N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.0000000000000001e135 or -8.09999999999999998e105 < im < -4.2e20 or 700 < im

   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 75.7%

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

   3. Taylor expanded in im around 0 58.3%

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

   4. Taylor expanded in im around inf 58.3%

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

      1. *-commutative 58.3%

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

      2. associate-*r* 58.3%

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

      3. *-commutative 58.3%

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

      4. associate-*r* 58.3%

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

   6. Simplified 58.3%

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

   if -6.0000000000000001e135 < im < -8.09999999999999998e105

   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%

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

      1. mul-1-neg 4.2%

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

      2. *-commutative 4.2%

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

      3. distribute-rgt-neg-in 4.2%

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

   4. Simplified 4.2%

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

   5. Taylor expanded in re around 0 37.1%

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

      1. +-commutative 37.1%

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

      2. mul-1-neg 37.1%

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

      3. unsub-neg 37.1%

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

      4. associate-*r* 37.1%

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

      5. distribute-rgt-out-- 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in re around inf 64.2%

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

      1. *-commutative 64.2%

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

      2. associate-*l* 64.2%

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

   10. Simplified 64.2%

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

   if -4.2e20 < im < 700

   1. Initial program 34.4%

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

   2. Taylor expanded in im around 0 95.0%

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

      1. mul-1-neg 95.0%

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

      2. *-commutative 95.0%

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

      3. distribute-rgt-neg-in 95.0%

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

   4. Simplified 95.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+135}:\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -8.1 \cdot 10^{+105}:\\ \;\;\;\;{re}^{3} \cdot \left(im \cdot 0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -4.2 \cdot 10^{+20} \lor \neg \left(im \leq 700\right):\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 11: 76.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{if}\;im \leq -6 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8.1 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{elif}\;im \leq -4.2 \cdot 10^{+20} \lor \neg \left(im \leq 480\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) (* re -0.16666666666666666))))
   (if (<= im -6e+135)
     t_0
     (if (<= im -8.1e+105)
       (* im (- (* 0.16666666666666666 (pow re 3.0)) re))
       (if (or (<= im -4.2e+20) (not (<= im 480.0)))
         t_0
         (* im (- (sin re))))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * (re * -0.16666666666666666);
	double tmp;
	if (im <= -6e+135) {
		tmp = t_0;
	} else if (im <= -8.1e+105) {
		tmp = im * ((0.16666666666666666 * pow(re, 3.0)) - re);
	} else if ((im <= -4.2e+20) || !(im <= 480.0)) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (re * (-0.16666666666666666d0))
    if (im <= (-6d+135)) then
        tmp = t_0
    else if (im <= (-8.1d+105)) then
        tmp = im * ((0.16666666666666666d0 * (re ** 3.0d0)) - re)
    else if ((im <= (-4.2d+20)) .or. (.not. (im <= 480.0d0))) then
        tmp = t_0
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * (re * -0.16666666666666666);
	double tmp;
	if (im <= -6e+135) {
		tmp = t_0;
	} else if (im <= -8.1e+105) {
		tmp = im * ((0.16666666666666666 * Math.pow(re, 3.0)) - re);
	} else if ((im <= -4.2e+20) || !(im <= 480.0)) {
		tmp = t_0;
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * (re * -0.16666666666666666)
	tmp = 0
	if im <= -6e+135:
		tmp = t_0
	elif im <= -8.1e+105:
		tmp = im * ((0.16666666666666666 * math.pow(re, 3.0)) - re)
	elif (im <= -4.2e+20) or not (im <= 480.0):
		tmp = t_0
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * Float64(re * -0.16666666666666666))
	tmp = 0.0
	if (im <= -6e+135)
		tmp = t_0;
	elseif (im <= -8.1e+105)
		tmp = Float64(im * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - re));
	elseif ((im <= -4.2e+20) || !(im <= 480.0))
		tmp = t_0;
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * (re * -0.16666666666666666);
	tmp = 0.0;
	if (im <= -6e+135)
		tmp = t_0;
	elseif (im <= -8.1e+105)
		tmp = im * ((0.16666666666666666 * (re ^ 3.0)) - re);
	elseif ((im <= -4.2e+20) || ~((im <= 480.0)))
		tmp = t_0;
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6e+135], t$95$0, If[LessEqual[im, -8.1e+105], N[(im * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -4.2e+20], N[Not[LessEqual[im, 480.0]], $MachinePrecision]], t$95$0, N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.0000000000000001e135 or -8.09999999999999998e105 < im < -4.2e20 or 480 < im

   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 75.7%

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

   3. Taylor expanded in im around 0 58.3%

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

   4. Taylor expanded in im around inf 58.3%

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

      1. *-commutative 58.3%

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

      2. associate-*r* 58.3%

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

      3. *-commutative 58.3%

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

      4. associate-*r* 58.3%

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

   6. Simplified 58.3%

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

   if -6.0000000000000001e135 < im < -8.09999999999999998e105

   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%

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

      1. mul-1-neg 4.2%

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

      2. *-commutative 4.2%

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

      3. distribute-rgt-neg-in 4.2%

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

   4. Simplified 4.2%

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

   5. Taylor expanded in re around 0 37.1%

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

      1. +-commutative 37.1%

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

      2. mul-1-neg 37.1%

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

      3. unsub-neg 37.1%

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

      4. associate-*r* 37.1%

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

      5. distribute-rgt-out-- 64.4%

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

   7. Simplified 64.4%

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

   if -4.2e20 < im < 480

   1. Initial program 34.4%

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

   2. Taylor expanded in im around 0 95.0%

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

      1. mul-1-neg 95.0%

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

      2. *-commutative 95.0%

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

      3. distribute-rgt-neg-in 95.0%

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

   4. Simplified 95.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+135}:\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;im \leq -8.1 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{elif}\;im \leq -4.2 \cdot 10^{+20} \lor \neg \left(im \leq 480\right):\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 12: 77.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.2 \cdot 10^{+20} \lor \neg \left(im \leq 820\right):\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.2e+20) (not (<= im 820.0)))
   (* (pow im 3.0) (* re -0.16666666666666666))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.2e+20) || !(im <= 820.0)) {
		tmp = pow(im, 3.0) * (re * -0.16666666666666666);
	} 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 <= (-3.2d+20)) .or. (.not. (im <= 820.0d0))) then
        tmp = (im ** 3.0d0) * (re * (-0.16666666666666666d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.2e+20) || !(im <= 820.0)) {
		tmp = Math.pow(im, 3.0) * (re * -0.16666666666666666);
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.2e+20) or not (im <= 820.0):
		tmp = math.pow(im, 3.0) * (re * -0.16666666666666666)
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.2e+20) || !(im <= 820.0))
		tmp = Float64((im ^ 3.0) * Float64(re * -0.16666666666666666));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.2e+20) || ~((im <= 820.0)))
		tmp = (im ^ 3.0) * (re * -0.16666666666666666);
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.2e+20], N[Not[LessEqual[im, 820.0]], $MachinePrecision]], N[(N[Power[im, 3.0], $MachinePrecision] * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.2e20 or 820 < im

   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 71.3%

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

   3. Taylor expanded in im around 0 55.5%

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

   4. Taylor expanded in im around inf 55.5%

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

      1. *-commutative 55.5%

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

      2. associate-*r* 55.5%

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

      3. *-commutative 55.5%

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

      4. associate-*r* 55.5%

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

   6. Simplified 55.5%

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

   if -3.2e20 < im < 820

   1. Initial program 34.4%

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

   2. Taylor expanded in im around 0 95.0%

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

      1. mul-1-neg 95.0%

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

      2. *-commutative 95.0%

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

      3. distribute-rgt-neg-in 95.0%

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

   4. Simplified 95.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.2 \cdot 10^{+20} \lor \neg \left(im \leq 820\right):\\ \;\;\;\;{im}^{3} \cdot \left(re \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 13: 58.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.3 \cdot 10^{+21} \lor \neg \left(im \leq 1.05 \cdot 10^{+129}\right):\\ \;\;\;\;-im \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.3e+21) (not (<= im 1.05e+129)))
   (- (* im re))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.3e+21) || !(im <= 1.05e+129)) {
		tmp = -(im * re);
	} 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 <= (-2.3d+21)) .or. (.not. (im <= 1.05d+129))) then
        tmp = -(im * re)
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.3e+21) || !(im <= 1.05e+129)) {
		tmp = -(im * re);
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.3e+21) or not (im <= 1.05e+129):
		tmp = -(im * re)
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.3e+21) || !(im <= 1.05e+129))
		tmp = Float64(-Float64(im * re));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.3e+21) || ~((im <= 1.05e+129)))
		tmp = -(im * re);
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.3e+21], N[Not[LessEqual[im, 1.05e+129]], $MachinePrecision]], (-N[(im * re), $MachinePrecision]), N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.3e21 or 1.04999999999999998e129 < 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 17.4%

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

      1. mul-1-neg 17.4%

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

      2. distribute-rgt-neg-in 17.4%

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

   7. Simplified 17.4%

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

   if -2.3e21 < im < 1.04999999999999998e129

   1. Initial program 44.0%

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

   2. Taylor expanded in im around 0 81.5%

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

      1. mul-1-neg 81.5%

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

      2. *-commutative 81.5%

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

      3. distribute-rgt-neg-in 81.5%

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

   4. Simplified 81.5%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.3 \cdot 10^{+21} \lor \neg \left(im \leq 1.05 \cdot 10^{+129}\right):\\ \;\;\;\;-im \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 14: 57.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.05 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{re \cdot \left(re \cdot 182.25\right)}\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+130}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -2.05e+33)
   (sqrt (* re (* re 182.25)))
   (if (<= im 1.06e+130) (* im (- (sin re))) (- (* im re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -2.05e+33) {
		tmp = sqrt((re * (re * 182.25)));
	} else if (im <= 1.06e+130) {
		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 <= (-2.05d+33)) then
        tmp = sqrt((re * (re * 182.25d0)))
    else if (im <= 1.06d+130) 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 <= -2.05e+33) {
		tmp = Math.sqrt((re * (re * 182.25)));
	} else if (im <= 1.06e+130) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = -(im * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -2.05e+33:
		tmp = math.sqrt((re * (re * 182.25)))
	elif im <= 1.06e+130:
		tmp = im * -math.sin(re)
	else:
		tmp = -(im * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -2.05e+33)
		tmp = sqrt(Float64(re * Float64(re * 182.25)));
	elseif (im <= 1.06e+130)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = Float64(-Float64(im * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -2.05e+33)
		tmp = sqrt((re * (re * 182.25)));
	elseif (im <= 1.06e+130)
		tmp = im * -sin(re);
	else
		tmp = -(im * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -2.05e+33], N[Sqrt[N[(re * N[(re * 182.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.06e+130], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], (-N[(im * re), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if im < -2.04999999999999997e33

   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 69.5%

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

   4. Applied egg-rr 2.3%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{27} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 23.1%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(0.5 \cdot re\right) \cdot 27\right)\right)} \]

      2. *-commutative 23.1%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(re \cdot 0.5\right)} \cdot 27\right)\right) \]

      3. associate-*l* 23.1%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{re \cdot \left(0.5 \cdot 27\right)}\right)\right) \]

      4. metadata-eval 23.1%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \color{blue}{13.5}\right)\right) \]

   6. Applied egg-rr 23.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot 13.5\right)\right)} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 2.3%

         \[\leadsto \color{blue}{re \cdot 13.5} \]

      2. add-sqr-sqrt 1.5%

         \[\leadsto \color{blue}{\sqrt{re \cdot 13.5} \cdot \sqrt{re \cdot 13.5}} \]

      3. sqrt-unprod 13.3%

         \[\leadsto \color{blue}{\sqrt{\left(re \cdot 13.5\right) \cdot \left(re \cdot 13.5\right)}} \]

      4. *-commutative 13.3%

         \[\leadsto \sqrt{\color{blue}{\left(13.5 \cdot re\right)} \cdot \left(re \cdot 13.5\right)} \]

      5. *-commutative 13.3%

         \[\leadsto \sqrt{\left(13.5 \cdot re\right) \cdot \color{blue}{\left(13.5 \cdot re\right)}} \]

      6. swap-sqr 13.3%

         \[\leadsto \sqrt{\color{blue}{\left(13.5 \cdot 13.5\right) \cdot \left(re \cdot re\right)}} \]

      7. metadata-eval 13.3%

         \[\leadsto \sqrt{\color{blue}{182.25} \cdot \left(re \cdot re\right)} \]

   8. Applied egg-rr 13.3%

      \[\leadsto \color{blue}{\sqrt{182.25 \cdot \left(re \cdot re\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 13.3%

         \[\leadsto \sqrt{\color{blue}{\left(182.25 \cdot re\right) \cdot re}} \]

   10. Simplified 13.3%

       \[\leadsto \color{blue}{\sqrt{\left(182.25 \cdot re\right) \cdot re}} \]

   if -2.04999999999999997e33 < im < 1.06e130

   1. Initial program 45.4%

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

   2. Taylor expanded in im around 0 79.5%

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

      1. mul-1-neg 79.5%

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

      2. *-commutative 79.5%

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

      3. distribute-rgt-neg-in 79.5%

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

   4. Simplified 79.5%

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

   if 1.06e130 < 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 5.3%

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

      1. mul-1-neg 5.3%

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

      2. *-commutative 5.3%

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

      3. distribute-rgt-neg-in 5.3%

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

   4. Simplified 5.3%

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

   5. Taylor expanded in re around 0 29.7%

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

      1. mul-1-neg 29.7%

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

      2. distribute-rgt-neg-in 29.7%

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

   7. Simplified 29.7%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.05 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{re \cdot \left(re \cdot 182.25\right)}\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+130}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-im \cdot re\\ \end{array} \]

Alternative 15: 33.7% accurate, 77.0× speedup

Math

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

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(-Float64(im * re))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

2. Taylor expanded in im around 0 51.7%

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

   1. mul-1-neg 51.7%

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

   2. *-commutative 51.7%

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

   3. distribute-rgt-neg-in 51.7%

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

4. Simplified 51.7%

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

5. Taylor expanded in re around 0 36.1%

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

   1. mul-1-neg 36.1%

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

   2. distribute-rgt-neg-in 36.1%

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

7. Simplified 36.1%

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

8. Final simplification 36.1%

   \[\leadsto -im \cdot re \]

Alternative 16: 2.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -3.0)

C

double code(double re, double im) {
	return -3.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -3.0d0
end function

Java

public static double code(double re, double im) {
	return -3.0;
}

Python

def code(re, im):
	return -3.0

Julia

function code(re, im)
	return -3.0
end

MATLAB

function tmp = code(re, im)
	tmp = -3.0;
end

Wolfram

code[re_, im_] := -3.0

Derivation

1. Initial program 65.7%

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

2. Taylor expanded in im around 0 51.7%

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

   1. mul-1-neg 51.7%

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

   2. *-commutative 51.7%

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

   3. distribute-rgt-neg-in 51.7%

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

4. Simplified 51.7%

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

6. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{-3} \]

7. Final simplification 2.7%

   \[\leadsto -3 \]

Alternative 17: 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 65.7%

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

2. Taylor expanded in im around 0 51.7%

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

   1. mul-1-neg 51.7%

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

   2. *-commutative 51.7%

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

   3. distribute-rgt-neg-in 51.7%

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

4. Simplified 51.7%

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

6. Applied egg-rr 2.8%

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

7. Final simplification 2.8%

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

Alternative 18: 15.3% 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 65.7%

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

2. Taylor expanded in im around 0 51.7%

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

   1. mul-1-neg 51.7%

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

   2. *-commutative 51.7%

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

   3. distribute-rgt-neg-in 51.7%

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

4. Simplified 51.7%

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

6. Applied egg-rr 15.0%

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

7. Final simplification 15.0%

   \[\leadsto 0 \]

Developer target: 99.7% 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 2023189 
(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))))