math.cos on complex, imaginary part

Percentage Accurate: 64.9% → 99.6%

Time: 9.9s

Alternatives: 15

Speedup: 2.8×

• 48.8% 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: 64.9% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := 0.5 \cdot \sin re\\ \mathbf{if}\;t_0 \leq -5 \lor \neg \left(t_0 \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* 0.5 (sin re))))
   (if (or (<= t_0 -5.0) (not (<= t_0 2e-12)))
     (* t_0 t_1)
     (*
      t_1
      (+
       (* im -2.0)
       (+
        (* -0.3333333333333333 (pow im 3.0))
        (* -0.016666666666666666 (pow im 5.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = 0.5 * sin(re);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 2e-12)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * pow(im, 3.0)) + (-0.016666666666666666 * pow(im, 5.0))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = 0.5d0 * sin(re)
    if ((t_0 <= (-5.0d0)) .or. (.not. (t_0 <= 2d-12))) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * ((im * (-2.0d0)) + (((-0.3333333333333333d0) * (im ** 3.0d0)) + ((-0.016666666666666666d0) * (im ** 5.0d0))))
    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 t_1 = 0.5 * Math.sin(re);
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 2e-12)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * Math.pow(im, 3.0)) + (-0.016666666666666666 * Math.pow(im, 5.0))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if (t_0 <= -5.0) or not (t_0 <= 2e-12):
		tmp = t_0 * t_1
	else:
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * math.pow(im, 3.0)) + (-0.016666666666666666 * math.pow(im, 5.0))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if ((t_0 <= -5.0) || !(t_0 <= 2e-12))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(Float64(im * -2.0) + Float64(Float64(-0.3333333333333333 * (im ^ 3.0)) + Float64(-0.016666666666666666 * (im ^ 5.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if ((t_0 <= -5.0) || ~((t_0 <= 2e-12)))
		tmp = t_0 * t_1;
	else
		tmp = t_1 * ((im * -2.0) + ((-0.3333333333333333 * (im ^ 3.0)) + (-0.016666666666666666 * (im ^ 5.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5.0], N[Not[LessEqual[t$95$0, 2e-12]], $MachinePrecision]], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -5 or 1.99999999999999996e-12 < (-.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 -5 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 1.99999999999999996e-12

   1. Initial program 29.5%

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

   2. Taylor expanded in im around 0 99.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.002 \lor \neg \left(t_0 \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.002) (not (<= t_0 2e-12)))
     (* 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 <= -0.002) || !(t_0 <= 2e-12)) {
		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 <= (-0.002d0)) .or. (.not. (t_0 <= 2d-12))) 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 <= -0.002) || !(t_0 <= 2e-12)) {
		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 <= -0.002) or not (t_0 <= 2e-12):
		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 <= -0.002) || !(t_0 <= 2e-12))
		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 <= -0.002) || ~((t_0 <= 2e-12)))
		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, -0.002], N[Not[LessEqual[t$95$0, 2e-12]], $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)) < -2e-3 or 1.99999999999999996e-12 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.9%

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

   if -2e-3 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 1.99999999999999996e-12

   1. Initial program 29.0%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-*r* 99.8%

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

      4. distribute-rgt-out 99.8%

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

      5. *-commutative 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in re around inf 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.002 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 93.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -2.2 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -130:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 5:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) (* (pow im 5.0) -0.008333333333333333))))
   (if (<= im -2.2e+70)
     t_0
     (if (<= im -130.0)
       (* (- (exp (- im)) (exp im)) (* 0.5 re))
       (if (<= im 5.0)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = sin(re) * (pow(im, 5.0) * -0.008333333333333333);
	double tmp;
	if (im <= -2.2e+70) {
		tmp = t_0;
	} else if (im <= -130.0) {
		tmp = (exp(-im) - exp(im)) * (0.5 * re);
	} else if (im <= 5.0) {
		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 = sin(re) * ((im ** 5.0d0) * (-0.008333333333333333d0))
    if (im <= (-2.2d+70)) then
        tmp = t_0
    else if (im <= (-130.0d0)) then
        tmp = (exp(-im) - exp(im)) * (0.5d0 * re)
    else if (im <= 5.0d0) 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.sin(re) * (Math.pow(im, 5.0) * -0.008333333333333333);
	double tmp;
	if (im <= -2.2e+70) {
		tmp = t_0;
	} else if (im <= -130.0) {
		tmp = (Math.exp(-im) - Math.exp(im)) * (0.5 * re);
	} else if (im <= 5.0) {
		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.sin(re) * (math.pow(im, 5.0) * -0.008333333333333333)
	tmp = 0
	if im <= -2.2e+70:
		tmp = t_0
	elif im <= -130.0:
		tmp = (math.exp(-im) - math.exp(im)) * (0.5 * re)
	elif im <= 5.0:
		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(sin(re) * Float64((im ^ 5.0) * -0.008333333333333333))
	tmp = 0.0
	if (im <= -2.2e+70)
		tmp = t_0;
	elseif (im <= -130.0)
		tmp = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 * re));
	elseif (im <= 5.0)
		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 = sin(re) * ((im ^ 5.0) * -0.008333333333333333);
	tmp = 0.0;
	if (im <= -2.2e+70)
		tmp = t_0;
	elseif (im <= -130.0)
		tmp = (exp(-im) - exp(im)) * (0.5 * re);
	elseif (im <= 5.0)
		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[Sin[re], $MachinePrecision] * N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.2e+70], t$95$0, If[LessEqual[im, -130.0], N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.0], 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 < -2.20000000000000001e70 or 5 < 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 93.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 93.8%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 93.8%

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

      2. *-commutative 93.8%

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

   5. Simplified 93.8%

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

   if -2.20000000000000001e70 < im < -130

   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 81.3%

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

      1. associate-*r* 81.3%

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

      2. *-commutative 81.3%

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

   4. Simplified 81.3%

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

   if -130 < im < 5

   1. Initial program 30.6%

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

   2. Taylor expanded in im around 0 98.5%

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

      1. associate-*r* 98.5%

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

      2. neg-mul-1 98.5%

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

      3. associate-*r* 98.5%

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

      4. distribute-rgt-out 98.5%

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

      5. *-commutative 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in re around inf 98.5%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.2 \cdot 10^{+70}:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{elif}\;im \leq -130:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 5:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \]

Alternative 4: 90.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5 \lor \neg \left(im \leq 5\right):\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -5.0) (not (<= im 5.0)))
   (* (sin re) (* (pow im 5.0) -0.008333333333333333))
   (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -5.0) || !(im <= 5.0)) {
		tmp = sin(re) * (pow(im, 5.0) * -0.008333333333333333);
	} 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) :: tmp
    if ((im <= (-5.0d0)) .or. (.not. (im <= 5.0d0))) then
        tmp = sin(re) * ((im ** 5.0d0) * (-0.008333333333333333d0))
    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 tmp;
	if ((im <= -5.0) || !(im <= 5.0)) {
		tmp = Math.sin(re) * (Math.pow(im, 5.0) * -0.008333333333333333);
	} else {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -5.0) or not (im <= 5.0):
		tmp = math.sin(re) * (math.pow(im, 5.0) * -0.008333333333333333)
	else:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -5.0) || !(im <= 5.0))
		tmp = Float64(sin(re) * Float64((im ^ 5.0) * -0.008333333333333333));
	else
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -5.0) || ~((im <= 5.0)))
		tmp = sin(re) * ((im ^ 5.0) * -0.008333333333333333);
	else
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -5.0], N[Not[LessEqual[im, 5.0]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] * N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $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 im < -5 or 5 < 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 83.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 83.8%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 83.8%

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

      2. *-commutative 83.8%

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

   5. Simplified 83.8%

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

   if -5 < im < 5

   1. Initial program 30.0%

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

   2. Taylor expanded in im around 0 99.2%

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

      1. associate-*r* 99.2%

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

      2. neg-mul-1 99.2%

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

      3. associate-*r* 99.2%

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

      4. distribute-rgt-out 99.2%

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

      5. *-commutative 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in re around inf 99.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5 \lor \neg \left(im \leq 5\right):\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 5: 89.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.3 \lor \neg \left(im \leq 3.3\right):\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.3) (not (<= im 3.3)))
   (* (sin re) (* (pow im 5.0) -0.008333333333333333))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.3) || !(im <= 3.3)) {
		tmp = sin(re) * (pow(im, 5.0) * -0.008333333333333333);
	} 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 <= (-3.3d0)) .or. (.not. (im <= 3.3d0))) then
        tmp = sin(re) * ((im ** 5.0d0) * (-0.008333333333333333d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.3) || !(im <= 3.3)) {
		tmp = Math.sin(re) * (Math.pow(im, 5.0) * -0.008333333333333333);
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.3) or not (im <= 3.3):
		tmp = math.sin(re) * (math.pow(im, 5.0) * -0.008333333333333333)
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.3) || !(im <= 3.3))
		tmp = Float64(sin(re) * Float64((im ^ 5.0) * -0.008333333333333333));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.3) || ~((im <= 3.3)))
		tmp = sin(re) * ((im ^ 5.0) * -0.008333333333333333);
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.3], N[Not[LessEqual[im, 3.3]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] * N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.2999999999999998 or 3.2999999999999998 < 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 83.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 83.8%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 83.8%

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

      2. *-commutative 83.8%

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

   5. Simplified 83.8%

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

   if -3.2999999999999998 < im < 3.2999999999999998

   1. Initial program 30.0%

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

   2. Taylor expanded in im around 0 98.5%

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

      1. associate-*r* 98.5%

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

      2. neg-mul-1 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.3 \lor \neg \left(im \leq 3.3\right):\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 6: 89.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 - im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (sin re) (- (* (pow im 5.0) -0.008333333333333333) im)))

C

double code(double re, double im) {
	return sin(re) * ((pow(im, 5.0) * -0.008333333333333333) - im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * (((im ** 5.0d0) * (-0.008333333333333333d0)) - im)
end function

Java

public static double code(double re, double im) {
	return Math.sin(re) * ((Math.pow(im, 5.0) * -0.008333333333333333) - im);
}

Python

def code(re, im):
	return math.sin(re) * ((math.pow(im, 5.0) * -0.008333333333333333) - im)

Julia

function code(re, im)
	return Float64(sin(re) * Float64(Float64((im ^ 5.0) * -0.008333333333333333) - im))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(re) * (((im ^ 5.0) * -0.008333333333333333) - im);
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

2. Taylor expanded in im around 0 91.8%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

3. Taylor expanded in im around inf 91.4%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-2 \cdot im + \color{blue}{-0.016666666666666666 \cdot {im}^{5}}\right) \]

4. Taylor expanded in im around 0 91.4%

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

   1. associate-*r* 91.4%

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

   2. neg-mul-1 91.4%

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

   3. associate-*r* 91.4%

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

   4. distribute-rgt-out 91.4%

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

   5. *-commutative 91.4%

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

6. Simplified 91.4%

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

7. Taylor expanded in re around inf 91.4%

   \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5} - im\right)} \]
8. Step-by-step derivation

   1. *-commutative 91.4%

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

9. Simplified 91.4%

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

10. Final simplification 91.4%

    \[\leadsto \sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 - im\right) \]

Alternative 7: 81.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.8 \cdot 10^{+51} \lor \neg \left(im \leq 8.2 \cdot 10^{+41}\right):\\ \;\;\;\;re \cdot \left({im}^{5} \cdot -0.008333333333333333 - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.8e+51) (not (<= im 8.2e+41)))
   (* re (- (* (pow im 5.0) -0.008333333333333333) im))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.8e+51) || !(im <= 8.2e+41)) {
		tmp = re * ((pow(im, 5.0) * -0.008333333333333333) - 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 <= (-4.8d+51)) .or. (.not. (im <= 8.2d+41))) then
        tmp = re * (((im ** 5.0d0) * (-0.008333333333333333d0)) - 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 <= -4.8e+51) || !(im <= 8.2e+41)) {
		tmp = re * ((Math.pow(im, 5.0) * -0.008333333333333333) - im);
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.8e+51) or not (im <= 8.2e+41):
		tmp = re * ((math.pow(im, 5.0) * -0.008333333333333333) - im)
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.8e+51) || !(im <= 8.2e+41))
		tmp = Float64(re * Float64(Float64((im ^ 5.0) * -0.008333333333333333) - im));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.8e+51) || ~((im <= 8.2e+41)))
		tmp = re * (((im ^ 5.0) * -0.008333333333333333) - im);
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.8e+51], N[Not[LessEqual[im, 8.2e+41]], $MachinePrecision]], N[(re * N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.7999999999999997e51 or 8.2000000000000007e41 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 97.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 97.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-2 \cdot im + \color{blue}{-0.016666666666666666 \cdot {im}^{5}}\right) \]

   4. Taylor expanded in im around 0 97.3%

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

      1. associate-*r* 97.3%

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

      2. neg-mul-1 97.3%

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

      3. associate-*r* 97.3%

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

      4. distribute-rgt-out 97.3%

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

      5. *-commutative 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in re around 0 69.6%

      \[\leadsto \color{blue}{re \cdot \left(-0.008333333333333333 \cdot {im}^{5} - im\right)} \]

   if -4.7999999999999997e51 < im < 8.2000000000000007e41

   1. Initial program 38.4%

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

   2. Taylor expanded in im around 0 87.2%

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

      1. associate-*r* 87.2%

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

      2. neg-mul-1 87.2%

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

   4. Simplified 87.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.8 \cdot 10^{+51} \lor \neg \left(im \leq 8.2 \cdot 10^{+41}\right):\\ \;\;\;\;re \cdot \left({im}^{5} \cdot -0.008333333333333333 - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 8: 76.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.8 \cdot 10^{+51}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+45}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -4.8e+51)
   (* -0.16666666666666666 (* re (pow im 3.0)))
   (if (<= im 2.5e+45)
     (* im (- (sin re)))
     (* re (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -4.8e+51) {
		tmp = -0.16666666666666666 * (re * pow(im, 3.0));
	} else if (im <= 2.5e+45) {
		tmp = im * -sin(re);
	} else {
		tmp = re * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-4.8d+51)) then
        tmp = (-0.16666666666666666d0) * (re * (im ** 3.0d0))
    else if (im <= 2.5d+45) then
        tmp = im * -sin(re)
    else
        tmp = re * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -4.8e+51) {
		tmp = -0.16666666666666666 * (re * Math.pow(im, 3.0));
	} else if (im <= 2.5e+45) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = re * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -4.8e+51:
		tmp = -0.16666666666666666 * (re * math.pow(im, 3.0))
	elif im <= 2.5e+45:
		tmp = im * -math.sin(re)
	else:
		tmp = re * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -4.8e+51)
		tmp = Float64(-0.16666666666666666 * Float64(re * (im ^ 3.0)));
	elseif (im <= 2.5e+45)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = Float64(re * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -4.8e+51)
		tmp = -0.16666666666666666 * (re * (im ^ 3.0));
	elseif (im <= 2.5e+45)
		tmp = im * -sin(re);
	else
		tmp = re * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -4.8e+51], N[(-0.16666666666666666 * N[(re * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+45], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(re * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.7999999999999997e51

   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 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

      3. associate-*r* 83.8%

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

      4. distribute-rgt-out 83.8%

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

      5. *-commutative 83.8%

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

   4. Simplified 83.8%

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

   5. Taylor expanded in re around 0 67.4%

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

   6. Taylor expanded in im around inf 67.4%

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

   if -4.7999999999999997e51 < im < 2.5e45

   1. Initial program 38.4%

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

   2. Taylor expanded in im around 0 87.2%

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

      1. associate-*r* 87.2%

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

      2. neg-mul-1 87.2%

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

   4. Simplified 87.2%

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

   if 2.5e45 < 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 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

      3. associate-*r* 83.8%

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

      4. distribute-rgt-out 83.8%

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

      5. *-commutative 83.8%

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

   4. Simplified 83.8%

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

   5. Taylor expanded in re around 0 56.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.8 \cdot 10^{+51}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+45}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 9: 76.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.8 \cdot 10^{+51} \lor \neg \left(im \leq 8.2 \cdot 10^{+41}\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.8e+51) (not (<= im 8.2e+41)))
   (* -0.16666666666666666 (* re (pow im 3.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.8e+51) || !(im <= 8.2e+41)) {
		tmp = -0.16666666666666666 * (re * pow(im, 3.0));
	} else {
		tmp = im * -sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-4.8d+51)) .or. (.not. (im <= 8.2d+41))) then
        tmp = (-0.16666666666666666d0) * (re * (im ** 3.0d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -4.8e+51) || !(im <= 8.2e+41)) {
		tmp = -0.16666666666666666 * (re * Math.pow(im, 3.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.8e+51) or not (im <= 8.2e+41):
		tmp = -0.16666666666666666 * (re * math.pow(im, 3.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.8e+51) || !(im <= 8.2e+41))
		tmp = Float64(-0.16666666666666666 * Float64(re * (im ^ 3.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.8e+51) || ~((im <= 8.2e+41)))
		tmp = -0.16666666666666666 * (re * (im ^ 3.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.8e+51], N[Not[LessEqual[im, 8.2e+41]], $MachinePrecision]], N[(-0.16666666666666666 * N[(re * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.7999999999999997e51 or 8.2000000000000007e41 < 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 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

      3. associate-*r* 83.8%

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

      4. distribute-rgt-out 83.8%

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

      5. *-commutative 83.8%

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

   4. Simplified 83.8%

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

   5. Taylor expanded in re around 0 62.3%

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

   6. Taylor expanded in im around inf 62.3%

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

   if -4.7999999999999997e51 < im < 8.2000000000000007e41

   1. Initial program 38.4%

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

   2. Taylor expanded in im around 0 87.2%

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

      1. associate-*r* 87.2%

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

      2. neg-mul-1 87.2%

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

   4. Simplified 87.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.8 \cdot 10^{+51} \lor \neg \left(im \leq 8.2 \cdot 10^{+41}\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 10: 57.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+54} \lor \neg \left(im \leq 1.12 \cdot 10^{+117}\right):\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.1e+54) (not (<= im 1.12e+117)))
   (* im (- re))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.1e+54) || !(im <= 1.12e+117)) {
		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 <= (-1.1d+54)) .or. (.not. (im <= 1.12d+117))) 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 <= -1.1e+54) || !(im <= 1.12e+117)) {
		tmp = im * -re;
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.1e+54) or not (im <= 1.12e+117):
		tmp = im * -re
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.1e+54) || !(im <= 1.12e+117))
		tmp = Float64(im * Float64(-re));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.1e+54) || ~((im <= 1.12e+117)))
		tmp = im * -re;
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.1e+54], N[Not[LessEqual[im, 1.12e+117]], $MachinePrecision]], N[(im * (-re)), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.09999999999999995e54 or 1.12000000000000002e117 < 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%

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

      1. associate-*r* 5.0%

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

      2. neg-mul-1 5.0%

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

   4. Simplified 5.0%

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

   5. Taylor expanded in re around 0 20.1%

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

      1. associate-*r* 20.1%

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

      2. neg-mul-1 20.1%

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

   7. Simplified 20.1%

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

   if -1.09999999999999995e54 < im < 1.12000000000000002e117

   1. Initial program 42.2%

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

   2. Taylor expanded in im around 0 82.0%

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

      1. associate-*r* 82.0%

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

      2. neg-mul-1 82.0%

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

   4. Simplified 82.0%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+54} \lor \neg \left(im \leq 1.12 \cdot 10^{+117}\right):\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 11: 33.5% 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 63.6%

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

2. Taylor expanded in im around 0 53.4%

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

   1. associate-*r* 53.4%

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

   2. neg-mul-1 53.4%

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

4. Simplified 53.4%

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

5. Taylor expanded in re around 0 33.3%

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

   1. associate-*r* 33.3%

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

   2. neg-mul-1 33.3%

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

7. Simplified 33.3%

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

8. Final simplification 33.3%

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

Alternative 12: 2.7% accurate, 308.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -512.0

Julia

function code(re, im)
	return -512.0
end

MATLAB

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

Wolfram

code[re_, im_] := -512.0

Derivation

1. Initial program 63.6%

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

2. Taylor expanded in im around 0 53.4%

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

   1. associate-*r* 53.4%

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

   2. neg-mul-1 53.4%

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

4. Simplified 53.4%

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

6. Applied egg-rr 2.7%

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

7. Final simplification 2.7%

   \[\leadsto -512 \]

Alternative 13: 2.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -0.004629629629629629 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -0.004629629629629629)

C

double code(double re, double im) {
	return -0.004629629629629629;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -0.004629629629629629d0
end function

Java

public static double code(double re, double im) {
	return -0.004629629629629629;
}

Python

def code(re, im):
	return -0.004629629629629629

Julia

function code(re, im)
	return -0.004629629629629629
end

MATLAB

function tmp = code(re, im)
	tmp = -0.004629629629629629;
end

Wolfram

code[re_, im_] := -0.004629629629629629

Derivation

1. Initial program 63.6%

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

2. Taylor expanded in im around 0 53.4%

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

   1. associate-*r* 53.4%

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

   2. neg-mul-1 53.4%

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

4. Simplified 53.4%

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

6. Applied egg-rr 2.8%

   \[\leadsto \color{blue}{-0.004629629629629629} \]

7. Final simplification 2.8%

   \[\leadsto -0.004629629629629629 \]

Alternative 14: 2.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -9.922903012752122 \cdot 10^{-17} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -9.922903012752122e-17)

C

double code(double re, double im) {
	return -9.922903012752122e-17;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -9.922903012752122d-17
end function

Java

public static double code(double re, double im) {
	return -9.922903012752122e-17;
}

Python

def code(re, im):
	return -9.922903012752122e-17

Julia

function code(re, im)
	return -9.922903012752122e-17
end

MATLAB

function tmp = code(re, im)
	tmp = -9.922903012752122e-17;
end

Wolfram

code[re_, im_] := -9.922903012752122e-17

Derivation

1. Initial program 63.6%

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

2. Taylor expanded in im around 0 53.4%

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

   1. associate-*r* 53.4%

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

   2. neg-mul-1 53.4%

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

4. Simplified 53.4%

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

6. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-9.922903012752122 \cdot 10^{-17}} \]

7. Final simplification 2.9%

   \[\leadsto -9.922903012752122 \cdot 10^{-17} \]

Alternative 15: 14.9% 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 63.6%

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

2. Taylor expanded in im around 0 53.4%

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

   1. associate-*r* 53.4%

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

   2. neg-mul-1 53.4%

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

4. Simplified 53.4%

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

6. Applied egg-rr 13.9%

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

7. Final simplification 13.9%

   \[\leadsto 0 \]

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 2023279 
(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))))