math.cos on complex, imaginary part

Percentage Accurate: 66.0% → 99.8%

Time: 7.9s

Alternatives: 14

Speedup: 2.8×

• Unsound rule application detected in e-graph. Results may not be sound.

• 50.0% 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: 66.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+23} \lor \neg \left(t_0 \leq 0.1\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{7} \cdot -0.0001984126984126984 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -4e+23) (not (<= t_0 0.1)))
     (* t_0 (* 0.5 (sin re)))
     (*
      (sin re)
      (+
       (* (pow im 5.0) -0.008333333333333333)
       (+
        (* (pow im 7.0) -0.0001984126984126984)
        (- (* (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 <= -4e+23) || !(t_0 <= 0.1)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im, 5.0) * -0.008333333333333333) + ((pow(im, 7.0) * -0.0001984126984126984) + ((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 <= (-4d+23)) .or. (.not. (t_0 <= 0.1d0))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im ** 5.0d0) * (-0.008333333333333333d0)) + (((im ** 7.0d0) * (-0.0001984126984126984d0)) + (((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 <= -4e+23) || !(t_0 <= 0.1)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im, 5.0) * -0.008333333333333333) + ((Math.pow(im, 7.0) * -0.0001984126984126984) + ((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 <= -4e+23) or not (t_0 <= 0.1):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im, 5.0) * -0.008333333333333333) + ((math.pow(im, 7.0) * -0.0001984126984126984) + ((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 <= -4e+23) || !(t_0 <= 0.1))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im ^ 5.0) * -0.008333333333333333) + Float64(Float64((im ^ 7.0) * -0.0001984126984126984) + 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 <= -4e+23) || ~((t_0 <= 0.1)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im ^ 5.0) * -0.008333333333333333) + (((im ^ 7.0) * -0.0001984126984126984) + (((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, -4e+23], N[Not[LessEqual[t$95$0, 0.1]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + N[(N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] + N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -3.9999999999999997e23 or 0.10000000000000001 < (-.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 -3.9999999999999997e23 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 0.10000000000000001

   1. Initial program 30.1%

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

   2. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]

      3. associate-+l+ 99.8%

         \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right)} \]

      4. associate-*r* 99.8%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \sin re\right) \cdot {im}^{5}} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      5. *-commutative 99.8%

         \[\leadsto \color{blue}{\left(\sin re \cdot -0.008333333333333333\right)} \cdot {im}^{5} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      6. associate-*l* 99.8%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      7. +-commutative 99.8%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      8. mul-1-neg 99.8%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \color{blue}{\left(-\sin re \cdot im\right)}\right) \]

      9. unsub-neg 99.8%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) - \sin re \cdot im\right)} \]

   4. Simplified 99.8%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{7} \cdot -0.0001984126984126984 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -4 \cdot 10^{+23} \lor \neg \left(e^{-im} - e^{im} \leq 0.1\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{7} \cdot -0.0001984126984126984 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+23} \lor \neg \left(t_0 \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -4e+23) (not (<= t_0 2e-9)))
     (* t_0 (* 0.5 (sin re)))
     (* (- im) (sin re)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -4e+23) || !(t_0 <= 2e-9)) {
		tmp = t_0 * (0.5 * sin(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) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-4d+23)) .or. (.not. (t_0 <= 2d-9))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = -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 <= -4e+23) || !(t_0 <= 2e-9)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -4e+23) or not (t_0 <= 2e-9):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = -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 <= -4e+23) || !(t_0 <= 2e-9))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(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 <= -4e+23) || ~((t_0 <= 2e-9)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = -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, -4e+23], N[Not[LessEqual[t$95$0, 2e-9]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -3.9999999999999997e23 or 2.00000000000000012e-9 < (-.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 -3.9999999999999997e23 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 2.00000000000000012e-9

   1. Initial program 29.6%

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

   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 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -4 \cdot 10^{+23} \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 3: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -2.45 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -19.5:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 5.6:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 7.0) (* (sin re) -0.0001984126984126984))))
   (if (<= im -2.45e+67)
     t_0
     (if (<= im -19.5)
       (* 0.5 (* (- (exp (- im)) (exp im)) re))
       (if (<= im 5.6)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = pow(im, 7.0) * (sin(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -2.45e+67) {
		tmp = t_0;
	} else if (im <= -19.5) {
		tmp = 0.5 * ((exp(-im) - exp(im)) * re);
	} else if (im <= 5.6) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im ** 7.0d0) * (sin(re) * (-0.0001984126984126984d0))
    if (im <= (-2.45d+67)) then
        tmp = t_0
    else if (im <= (-19.5d0)) then
        tmp = 0.5d0 * ((exp(-im) - exp(im)) * re)
    else if (im <= 5.6d0) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 7.0) * (Math.sin(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -2.45e+67) {
		tmp = t_0;
	} else if (im <= -19.5) {
		tmp = 0.5 * ((Math.exp(-im) - Math.exp(im)) * re);
	} else if (im <= 5.6) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 7.0) * (math.sin(re) * -0.0001984126984126984)
	tmp = 0
	if im <= -2.45e+67:
		tmp = t_0
	elif im <= -19.5:
		tmp = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	elif im <= 5.6:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 7.0) * Float64(sin(re) * -0.0001984126984126984))
	tmp = 0.0
	if (im <= -2.45e+67)
		tmp = t_0;
	elseif (im <= -19.5)
		tmp = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re));
	elseif (im <= 5.6)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 7.0) * (sin(re) * -0.0001984126984126984);
	tmp = 0.0;
	if (im <= -2.45e+67)
		tmp = t_0;
	elseif (im <= -19.5)
		tmp = 0.5 * ((exp(-im) - exp(im)) * re);
	elseif (im <= 5.6)
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 7.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.45e+67], t$95$0, If[LessEqual[im, -19.5], N[(0.5 * N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.44999999999999995e67 or 5.5999999999999996 < 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 94.9%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 94.9%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      2. +-commutative 94.9%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]

      3. associate-+l+ 94.9%

         \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right)} \]

      4. associate-*r* 94.9%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \sin re\right) \cdot {im}^{5}} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      5. *-commutative 94.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot -0.008333333333333333\right)} \cdot {im}^{5} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      6. associate-*l* 94.9%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      7. +-commutative 94.9%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      8. mul-1-neg 94.9%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \color{blue}{\left(-\sin re \cdot im\right)}\right) \]

      9. unsub-neg 94.9%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) - \sin re \cdot im\right)} \]

   4. Simplified 94.9%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{7} \cdot -0.0001984126984126984 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\right)} \]

   5. Taylor expanded in im around inf 94.9%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 94.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} \]

      2. *-commutative 94.9%

         \[\leadsto \color{blue}{\left({im}^{7} \cdot \sin re\right)} \cdot -0.0001984126984126984 \]

      3. associate-*l* 94.9%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   7. Simplified 94.9%

      \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   if -2.44999999999999995e67 < im < -19.5

   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 62.4%

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

   if -19.5 < im < 5.5999999999999996

   1. Initial program 30.7%

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

   2. Taylor expanded in im around 0 98.8%

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

      1. mul-1-neg 98.8%

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

      2. unsub-neg 98.8%

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

      3. *-commutative 98.8%

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

      4. associate-*l* 98.8%

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

      5. distribute-lft-out-- 98.8%

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

   4. Simplified 98.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.45 \cdot 10^{+67}:\\ \;\;\;\;{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq -19.5:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 5.6:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 4: 94.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -2.45 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -19.5:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 4.2:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 7.0) (* (sin re) -0.0001984126984126984))))
   (if (<= im -2.45e+67)
     t_0
     (if (<= im -19.5)
       (* 0.5 (* (- (exp (- im)) (exp im)) re))
       (if (<= im 4.2) (* (- im) (sin re)) t_0)))))

C

double code(double re, double im) {
	double t_0 = pow(im, 7.0) * (sin(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -2.45e+67) {
		tmp = t_0;
	} else if (im <= -19.5) {
		tmp = 0.5 * ((exp(-im) - exp(im)) * re);
	} else if (im <= 4.2) {
		tmp = -im * sin(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im ** 7.0d0) * (sin(re) * (-0.0001984126984126984d0))
    if (im <= (-2.45d+67)) then
        tmp = t_0
    else if (im <= (-19.5d0)) then
        tmp = 0.5d0 * ((exp(-im) - exp(im)) * re)
    else if (im <= 4.2d0) then
        tmp = -im * sin(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 7.0) * (Math.sin(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -2.45e+67) {
		tmp = t_0;
	} else if (im <= -19.5) {
		tmp = 0.5 * ((Math.exp(-im) - Math.exp(im)) * re);
	} else if (im <= 4.2) {
		tmp = -im * Math.sin(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 7.0) * (math.sin(re) * -0.0001984126984126984)
	tmp = 0
	if im <= -2.45e+67:
		tmp = t_0
	elif im <= -19.5:
		tmp = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	elif im <= 4.2:
		tmp = -im * math.sin(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 7.0) * Float64(sin(re) * -0.0001984126984126984))
	tmp = 0.0
	if (im <= -2.45e+67)
		tmp = t_0;
	elseif (im <= -19.5)
		tmp = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re));
	elseif (im <= 4.2)
		tmp = Float64(Float64(-im) * sin(re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 7.0) * (sin(re) * -0.0001984126984126984);
	tmp = 0.0;
	if (im <= -2.45e+67)
		tmp = t_0;
	elseif (im <= -19.5)
		tmp = 0.5 * ((exp(-im) - exp(im)) * re);
	elseif (im <= 4.2)
		tmp = -im * sin(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 7.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.45e+67], t$95$0, If[LessEqual[im, -19.5], N[(0.5 * N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.44999999999999995e67 or 4.20000000000000018 < 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 94.9%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 94.9%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      2. +-commutative 94.9%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]

      3. associate-+l+ 94.9%

         \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right)} \]

      4. associate-*r* 94.9%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \sin re\right) \cdot {im}^{5}} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      5. *-commutative 94.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot -0.008333333333333333\right)} \cdot {im}^{5} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      6. associate-*l* 94.9%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      7. +-commutative 94.9%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      8. mul-1-neg 94.9%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \color{blue}{\left(-\sin re \cdot im\right)}\right) \]

      9. unsub-neg 94.9%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) - \sin re \cdot im\right)} \]

   4. Simplified 94.9%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{7} \cdot -0.0001984126984126984 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\right)} \]

   5. Taylor expanded in im around inf 94.9%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 94.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} \]

      2. *-commutative 94.9%

         \[\leadsto \color{blue}{\left({im}^{7} \cdot \sin re\right)} \cdot -0.0001984126984126984 \]

      3. associate-*l* 94.9%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   7. Simplified 94.9%

      \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   if -2.44999999999999995e67 < im < -19.5

   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 62.4%

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

   if -19.5 < im < 4.20000000000000018

   1. Initial program 30.7%

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

   2. Taylor expanded in im around 0 98.6%

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

      1. mul-1-neg 98.6%

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

      2. *-commutative 98.6%

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

      3. distribute-rgt-neg-in 98.6%

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

   4. Simplified 98.6%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.45 \cdot 10^{+67}:\\ \;\;\;\;{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq -19.5:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 4.2:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 5: 92.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.2 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.2) (not (<= im 4.2)))
   (* (pow im 7.0) (* (sin re) -0.0001984126984126984))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.2) || !(im <= 4.2)) {
		tmp = pow(im, 7.0) * (sin(re) * -0.0001984126984126984);
	} 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 <= (-4.2d0)) .or. (.not. (im <= 4.2d0))) then
        tmp = (im ** 7.0d0) * (sin(re) * (-0.0001984126984126984d0))
    else
        tmp = -im * sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -4.2) || !(im <= 4.2)) {
		tmp = Math.pow(im, 7.0) * (Math.sin(re) * -0.0001984126984126984);
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.2) or not (im <= 4.2):
		tmp = math.pow(im, 7.0) * (math.sin(re) * -0.0001984126984126984)
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.2) || !(im <= 4.2))
		tmp = Float64((im ^ 7.0) * Float64(sin(re) * -0.0001984126984126984));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.2) || ~((im <= 4.2)))
		tmp = (im ^ 7.0) * (sin(re) * -0.0001984126984126984);
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.2], N[Not[LessEqual[im, 4.2]], $MachinePrecision]], N[(N[Power[im, 7.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.20000000000000018 or 4.20000000000000018 < 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 85.4%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 85.4%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      2. +-commutative 85.4%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]

      3. associate-+l+ 85.4%

         \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right)} \]

      4. associate-*r* 85.4%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \sin re\right) \cdot {im}^{5}} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      5. *-commutative 85.4%

         \[\leadsto \color{blue}{\left(\sin re \cdot -0.008333333333333333\right)} \cdot {im}^{5} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      6. associate-*l* 85.4%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      7. +-commutative 85.4%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      8. mul-1-neg 85.4%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \color{blue}{\left(-\sin re \cdot im\right)}\right) \]

      9. unsub-neg 85.4%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) - \sin re \cdot im\right)} \]

   4. Simplified 85.4%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{7} \cdot -0.0001984126984126984 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\right)} \]

   5. Taylor expanded in im around inf 85.4%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 85.4%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} \]

      2. *-commutative 85.4%

         \[\leadsto \color{blue}{\left({im}^{7} \cdot \sin re\right)} \cdot -0.0001984126984126984 \]

      3. associate-*l* 85.4%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   7. Simplified 85.4%

      \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   if -4.20000000000000018 < im < 4.20000000000000018

   1. Initial program 30.1%

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

   2. Taylor expanded in im around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.2 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 6: 59.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{if}\;im \leq -3.4 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 580:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 6.4 \cdot 10^{+125}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.16666666666666666 (* im (pow re 3.0)))))
   (if (<= im -3.4e+14)
     t_0
     (if (<= im 580.0)
       (* (- im) (sin re))
       (if (<= im 6.4e+125) t_0 (* im (- re)))))))

C

double code(double re, double im) {
	double t_0 = 0.16666666666666666 * (im * pow(re, 3.0));
	double tmp;
	if (im <= -3.4e+14) {
		tmp = t_0;
	} else if (im <= 580.0) {
		tmp = -im * sin(re);
	} else if (im <= 6.4e+125) {
		tmp = t_0;
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 * (im * (re ** 3.0d0))
    if (im <= (-3.4d+14)) then
        tmp = t_0
    else if (im <= 580.0d0) then
        tmp = -im * sin(re)
    else if (im <= 6.4d+125) then
        tmp = t_0
    else
        tmp = im * -re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.16666666666666666 * (im * Math.pow(re, 3.0));
	double tmp;
	if (im <= -3.4e+14) {
		tmp = t_0;
	} else if (im <= 580.0) {
		tmp = -im * Math.sin(re);
	} else if (im <= 6.4e+125) {
		tmp = t_0;
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.16666666666666666 * (im * math.pow(re, 3.0))
	tmp = 0
	if im <= -3.4e+14:
		tmp = t_0
	elif im <= 580.0:
		tmp = -im * math.sin(re)
	elif im <= 6.4e+125:
		tmp = t_0
	else:
		tmp = im * -re
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.16666666666666666 * Float64(im * (re ^ 3.0)))
	tmp = 0.0
	if (im <= -3.4e+14)
		tmp = t_0;
	elseif (im <= 580.0)
		tmp = Float64(Float64(-im) * sin(re));
	elseif (im <= 6.4e+125)
		tmp = t_0;
	else
		tmp = Float64(im * Float64(-re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.16666666666666666 * (im * (re ^ 3.0));
	tmp = 0.0;
	if (im <= -3.4e+14)
		tmp = t_0;
	elseif (im <= 580.0)
		tmp = -im * sin(re);
	elseif (im <= 6.4e+125)
		tmp = t_0;
	else
		tmp = im * -re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(im * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.4e+14], t$95$0, If[LessEqual[im, 580.0], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.4e+125], t$95$0, N[(im * (-re)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.4e14 or 580 < im < 6.39999999999999967e125

   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 4.3%

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

      1. mul-1-neg 4.3%

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

      2. *-commutative 4.3%

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

      3. distribute-rgt-neg-in 4.3%

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

   4. Simplified 4.3%

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

   5. Taylor expanded in re around 0 14.0%

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

      1. associate-*r* 14.0%

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

      2. associate-*r* 14.0%

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

      3. distribute-rgt-out 31.6%

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

      4. mul-1-neg 31.6%

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

   7. Simplified 31.6%

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

   8. Taylor expanded in re around inf 30.7%

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

   if -3.4e14 < im < 580

   1. Initial program 33.3%

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

   2. Taylor expanded in im around 0 95.1%

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

      1. mul-1-neg 95.1%

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

      2. *-commutative 95.1%

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

      3. distribute-rgt-neg-in 95.1%

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

   4. Simplified 95.1%

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

   if 6.39999999999999967e125 < 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 5.0%

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

      1. mul-1-neg 5.0%

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

      2. *-commutative 5.0%

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

      3. distribute-rgt-neg-in 5.0%

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

   4. Simplified 5.0%

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

   5. Taylor expanded in re around 0 24.4%

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

      1. mul-1-neg 24.4%

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

      2. distribute-rgt-neg-in 24.4%

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

   7. Simplified 24.4%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.4 \cdot 10^{+14}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{elif}\;im \leq 580:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 6.4 \cdot 10^{+125}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]

Alternative 7: 81.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -22000000000000 \lor \neg \left(im \leq 9.4 \cdot 10^{+29}\right):\\ \;\;\;\;{im}^{7} \cdot \left(re \cdot -0.0001984126984126984\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -22000000000000.0) (not (<= im 9.4e+29)))
   (* (pow im 7.0) (* re -0.0001984126984126984))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -22000000000000.0) || !(im <= 9.4e+29)) {
		tmp = pow(im, 7.0) * (re * -0.0001984126984126984);
	} 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 <= (-22000000000000.0d0)) .or. (.not. (im <= 9.4d+29))) then
        tmp = (im ** 7.0d0) * (re * (-0.0001984126984126984d0))
    else
        tmp = -im * sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -22000000000000.0) || !(im <= 9.4e+29)) {
		tmp = Math.pow(im, 7.0) * (re * -0.0001984126984126984);
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -22000000000000.0) or not (im <= 9.4e+29):
		tmp = math.pow(im, 7.0) * (re * -0.0001984126984126984)
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -22000000000000.0) || !(im <= 9.4e+29))
		tmp = Float64((im ^ 7.0) * Float64(re * -0.0001984126984126984));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -22000000000000.0) || ~((im <= 9.4e+29)))
		tmp = (im ^ 7.0) * (re * -0.0001984126984126984);
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -22000000000000.0], N[Not[LessEqual[im, 9.4e+29]], $MachinePrecision]], N[(N[Power[im, 7.0], $MachinePrecision] * N[(re * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.2e13 or 9.4000000000000004e29 < 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 91.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 91.3%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      2. +-commutative 91.3%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]

      3. associate-+l+ 91.3%

         \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right)} \]

      4. associate-*r* 91.3%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \sin re\right) \cdot {im}^{5}} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      5. *-commutative 91.3%

         \[\leadsto \color{blue}{\left(\sin re \cdot -0.008333333333333333\right)} \cdot {im}^{5} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      6. associate-*l* 91.3%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)} + \left(-1 \cdot \left(\sin re \cdot im\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)\right) \]

      7. +-commutative 91.3%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      8. mul-1-neg 91.3%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \color{blue}{\left(-\sin re \cdot im\right)}\right) \]

      9. unsub-neg 91.3%

         \[\leadsto \sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\left(\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) - \sin re \cdot im\right)} \]

   4. Simplified 91.3%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{7} \cdot -0.0001984126984126984 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\right)} \]

   5. Taylor expanded in im around inf 91.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 91.3%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} \]

      2. *-commutative 91.3%

         \[\leadsto \color{blue}{\left({im}^{7} \cdot \sin re\right)} \cdot -0.0001984126984126984 \]

      3. associate-*l* 91.3%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   7. Simplified 91.3%

      \[\leadsto \color{blue}{{im}^{7} \cdot \left(\sin re \cdot -0.0001984126984126984\right)} \]

   8. Taylor expanded in re around 0 66.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 66.0%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot re\right) \cdot {im}^{7}} \]

      2. *-commutative 66.0%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(-0.0001984126984126984 \cdot re\right)} \]

   10. Simplified 66.0%

       \[\leadsto \color{blue}{{im}^{7} \cdot \left(-0.0001984126984126984 \cdot re\right)} \]

   if -2.2e13 < im < 9.4000000000000004e29

   1. Initial program 34.7%

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

   2. Taylor expanded in im around 0 93.1%

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

      1. mul-1-neg 93.1%

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

      2. *-commutative 93.1%

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

      3. distribute-rgt-neg-in 93.1%

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

   4. Simplified 93.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -22000000000000 \lor \neg \left(im \leq 9.4 \cdot 10^{+29}\right):\\ \;\;\;\;{im}^{7} \cdot \left(re \cdot -0.0001984126984126984\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 8: 55.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6.4 \cdot 10^{+103} \lor \neg \left(im \leq 5.9 \cdot 10^{+134}\right):\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -6.4e+103) (not (<= im 5.9e+134)))
   (* im (- re))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -6.4e+103) || !(im <= 5.9e+134)) {
		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 <= (-6.4d+103)) .or. (.not. (im <= 5.9d+134))) 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 <= -6.4e+103) || !(im <= 5.9e+134)) {
		tmp = im * -re;
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -6.4e+103) or not (im <= 5.9e+134):
		tmp = im * -re
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -6.4e+103) || !(im <= 5.9e+134))
		tmp = Float64(im * Float64(-re));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -6.4e+103) || ~((im <= 5.9e+134)))
		tmp = im * -re;
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -6.4e+103], N[Not[LessEqual[im, 5.9e+134]], $MachinePrecision]], N[(im * (-re)), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -6.39999999999999985e103 or 5.90000000000000008e134 < 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 5.2%

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

      1. mul-1-neg 5.2%

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

      2. *-commutative 5.2%

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

      3. distribute-rgt-neg-in 5.2%

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

   4. Simplified 5.2%

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

   5. Taylor expanded in re around 0 20.5%

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

      1. mul-1-neg 20.5%

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

      2. distribute-rgt-neg-in 20.5%

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

   7. Simplified 20.5%

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

   if -6.39999999999999985e103 < im < 5.90000000000000008e134

   1. Initial program 50.4%

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

   2. Taylor expanded in im around 0 71.5%

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

      1. mul-1-neg 71.5%

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

      2. *-commutative 71.5%

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

      3. distribute-rgt-neg-in 71.5%

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

   4. Simplified 71.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.4 \cdot 10^{+103} \lor \neg \left(im \leq 5.9 \cdot 10^{+134}\right):\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 9: 32.7% accurate, 77.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

2. Taylor expanded in im around 0 51.6%

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

   1. mul-1-neg 51.6%

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

   2. *-commutative 51.6%

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

   3. distribute-rgt-neg-in 51.6%

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

4. Simplified 51.6%

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

5. Taylor expanded in re around 0 30.2%

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

   1. mul-1-neg 30.2%

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

   2. distribute-rgt-neg-in 30.2%

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

7. Simplified 30.2%

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

8. Final simplification 30.2%

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

Alternative 10: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.5)

C

double code(double re, double im) {
	return -1.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -1.5d0
end function

Java

public static double code(double re, double im) {
	return -1.5;
}

Python

def code(re, im):
	return -1.5

Julia

function code(re, im)
	return -1.5
end

MATLAB

function tmp = code(re, im)
	tmp = -1.5;
end

Wolfram

code[re_, im_] := -1.5

Derivation

1. Initial program 65.3%

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

2. Taylor expanded in re around 0 48.6%

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

4. Applied egg-rr 2.8%

   \[\leadsto 0.5 \cdot \color{blue}{-3} \]

5. Final simplification 2.8%

   \[\leadsto -1.5 \]

Alternative 11: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -0.004166666666666667 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -0.004166666666666667)

C

double code(double re, double im) {
	return -0.004166666666666667;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -0.004166666666666667d0
end function

Java

public static double code(double re, double im) {
	return -0.004166666666666667;
}

Python

def code(re, im):
	return -0.004166666666666667

Julia

function code(re, im)
	return -0.004166666666666667
end

MATLAB

function tmp = code(re, im)
	tmp = -0.004166666666666667;
end

Wolfram

code[re_, im_] := -0.004166666666666667

Derivation

1. Initial program 65.3%

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

2. Taylor expanded in re around 0 48.6%

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

4. Applied egg-rr 2.8%

   \[\leadsto 0.5 \cdot \color{blue}{-0.008333333333333333} \]

5. Final simplification 2.8%

   \[\leadsto -0.004166666666666667 \]

Alternative 12: 2.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -3.9055157630365495 \cdot 10^{-12} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -3.9055157630365495e-12)

C

double code(double re, double im) {
	return -3.9055157630365495e-12;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -3.9055157630365495d-12
end function

Java

public static double code(double re, double im) {
	return -3.9055157630365495e-12;
}

Python

def code(re, im):
	return -3.9055157630365495e-12

Julia

function code(re, im)
	return -3.9055157630365495e-12
end

MATLAB

function tmp = code(re, im)
	tmp = -3.9055157630365495e-12;
end

Wolfram

code[re_, im_] := -3.9055157630365495e-12

Derivation

1. Initial program 65.3%

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

2. Taylor expanded in re around 0 48.6%

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

4. Applied egg-rr 2.8%

   \[\leadsto 0.5 \cdot \color{blue}{-7.811031526073099 \cdot 10^{-12}} \]

5. Final simplification 2.8%

   \[\leadsto -3.9055157630365495 \cdot 10^{-12} \]

Alternative 13: 2.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -2.382841615671091 \cdot 10^{-34} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -2.382841615671091e-34)

C

double code(double re, double im) {
	return -2.382841615671091e-34;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -2.382841615671091d-34
end function

Java

public static double code(double re, double im) {
	return -2.382841615671091e-34;
}

Python

def code(re, im):
	return -2.382841615671091e-34

Julia

function code(re, im)
	return -2.382841615671091e-34
end

MATLAB

function tmp = code(re, im)
	tmp = -2.382841615671091e-34;
end

Wolfram

code[re_, im_] := -2.382841615671091e-34

Derivation

1. Initial program 65.3%

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

2. Taylor expanded in re around 0 48.6%

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

4. Applied egg-rr 2.9%

   \[\leadsto 0.5 \cdot \color{blue}{-4.765683231342182 \cdot 10^{-34}} \]

5. Final simplification 2.9%

   \[\leadsto -2.382841615671091 \cdot 10^{-34} \]

Alternative 14: 14.8% 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 65.3%

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

2. Taylor expanded in re around 0 48.6%

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

4. Applied egg-rr 14.6%

   \[\leadsto 0.5 \cdot \color{blue}{0} \]

5. Final simplification 14.6%

   \[\leadsto 0 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023178 
(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))))