math.sin on complex, imaginary part

Percentage Accurate: 54.6% → 99.8%

Time: 8.6s

Alternatives: 14

Speedup: 2.9×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 54.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.01 \lor \neg \left(t_0 \leq 10^{-6}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.01) (not (<= t_0 1e-6)))
     (* (* (cos re) 0.5) t_0)
     (* (cos 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 <= -0.01) || !(t_0 <= 1e-6)) {
		tmp = (cos(re) * 0.5) * t_0;
	} else {
		tmp = cos(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 <= (-0.01d0)) .or. (.not. (t_0 <= 1d-6))) then
        tmp = (cos(re) * 0.5d0) * t_0
    else
        tmp = cos(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 <= -0.01) || !(t_0 <= 1e-6)) {
		tmp = (Math.cos(re) * 0.5) * t_0;
	} else {
		tmp = Math.cos(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 <= -0.01) or not (t_0 <= 1e-6):
		tmp = (math.cos(re) * 0.5) * t_0
	else:
		tmp = math.cos(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 <= -0.01) || !(t_0 <= 1e-6))
		tmp = Float64(Float64(cos(re) * 0.5) * t_0);
	else
		tmp = Float64(cos(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 <= -0.01) || ~((t_0 <= 1e-6)))
		tmp = (cos(re) * 0.5) * t_0;
	else
		tmp = cos(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, -0.01], N[Not[LessEqual[t$95$0, 1e-6]], $MachinePrecision]], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[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 (-.f64 0 im)) (exp.f64 im)) < -0.0100000000000000002 or 9.99999999999999955e-7 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   if -0.0100000000000000002 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 9.99999999999999955e-7

   1. Initial program 7.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 7.7%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. distribute-lft-out-- 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.01 \lor \neg \left(e^{-im} - e^{im} \leq 10^{-6}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq 0.005:\\ \;\;\;\;\cos re \cdot \left(\mathsf{fma}\left({im}^{5}, -0.008333333333333333, -im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (<= t_0 0.005)
     (*
      (cos re)
      (+
       (fma (pow im 5.0) -0.008333333333333333 (- im))
       (+
        (* (pow im 7.0) -0.0001984126984126984)
        (* (pow im 3.0) -0.16666666666666666))))
     (* (* (cos re) 0.5) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if (t_0 <= 0.005) {
		tmp = cos(re) * (fma(pow(im, 5.0), -0.008333333333333333, -im) + ((pow(im, 7.0) * -0.0001984126984126984) + (pow(im, 3.0) * -0.16666666666666666)));
	} else {
		tmp = (cos(re) * 0.5) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if (t_0 <= 0.005)
		tmp = Float64(cos(re) * Float64(fma((im ^ 5.0), -0.008333333333333333, Float64(-im)) + Float64(Float64((im ^ 7.0) * -0.0001984126984126984) + Float64((im ^ 3.0) * -0.16666666666666666))));
	else
		tmp = Float64(Float64(cos(re) * 0.5) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.005], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333 + (-im)), $MachinePrecision] + N[(N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] + N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.0050000000000000001

   1. Initial program 35.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 35.3%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in im around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. associate-+r+ 98.5%

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

      3. associate-+l+ 98.5%

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

      4. mul-1-neg 98.5%

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

      5. distribute-rgt-neg-in 98.5%

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

      6. *-commutative 98.5%

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

      7. associate-*l* 98.5%

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

      8. distribute-lft-out 98.5%

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

      9. *-commutative 98.5%

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

      10. associate-*r* 98.5%

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

      11. *-commutative 98.5%

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

      12. associate-*r* 98.5%

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

   6. Simplified 98.5%

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

   if 0.0050000000000000001 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq 0.005:\\ \;\;\;\;\cos re \cdot \left(\mathsf{fma}\left({im}^{5}, -0.008333333333333333, -im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 3: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := \cos re \cdot 0.5\\ \mathbf{if}\;t_0 \leq 0.005:\\ \;\;\;\;t_1 \cdot \left(im \cdot -2 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* (cos re) 0.5)))
   (if (<= t_0 0.005)
     (*
      t_1
      (+
       (* im -2.0)
       (+
        (* (pow im 5.0) -0.016666666666666666)
        (* (pow im 3.0) -0.3333333333333333))))
     (* t_1 t_0))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = cos(re) * 0.5;
	double tmp;
	if (t_0 <= 0.005) {
		tmp = t_1 * ((im * -2.0) + ((pow(im, 5.0) * -0.016666666666666666) + (pow(im, 3.0) * -0.3333333333333333)));
	} else {
		tmp = t_1 * 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) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = cos(re) * 0.5d0
    if (t_0 <= 0.005d0) then
        tmp = t_1 * ((im * (-2.0d0)) + (((im ** 5.0d0) * (-0.016666666666666666d0)) + ((im ** 3.0d0) * (-0.3333333333333333d0))))
    else
        tmp = t_1 * t_0
    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 = Math.cos(re) * 0.5;
	double tmp;
	if (t_0 <= 0.005) {
		tmp = t_1 * ((im * -2.0) + ((Math.pow(im, 5.0) * -0.016666666666666666) + (Math.pow(im, 3.0) * -0.3333333333333333)));
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.cos(re) * 0.5
	tmp = 0
	if t_0 <= 0.005:
		tmp = t_1 * ((im * -2.0) + ((math.pow(im, 5.0) * -0.016666666666666666) + (math.pow(im, 3.0) * -0.3333333333333333)))
	else:
		tmp = t_1 * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(cos(re) * 0.5)
	tmp = 0.0
	if (t_0 <= 0.005)
		tmp = Float64(t_1 * Float64(Float64(im * -2.0) + Float64(Float64((im ^ 5.0) * -0.016666666666666666) + Float64((im ^ 3.0) * -0.3333333333333333))));
	else
		tmp = Float64(t_1 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = cos(re) * 0.5;
	tmp = 0.0;
	if (t_0 <= 0.005)
		tmp = t_1 * ((im * -2.0) + (((im ^ 5.0) * -0.016666666666666666) + ((im ^ 3.0) * -0.3333333333333333)));
	else
		tmp = t_1 * t_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[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, 0.005], N[(t$95$1 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision] + N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.0050000000000000001

   1. Initial program 35.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 35.3%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in im around 0 98.0%

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

   if 0.0050000000000000001 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq 0.005:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot -2 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 4: 86.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ t_1 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_2 := t_0 - im\\ t_3 := t_2 \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{if}\;im \leq -2 \cdot 10^{+238}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2 \cdot 10^{+167}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq -0.00098:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 0.00115:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+280}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666))
        (t_1 (* 0.5 (- (exp (- im)) (exp im))))
        (t_2 (- t_0 im))
        (t_3 (* t_2 (+ (* -0.5 (* re re)) 1.0))))
   (if (<= im -2e+238)
     t_0
     (if (<= im -2e+167)
       t_3
       (if (<= im -0.00098)
         t_1
         (if (<= im 0.00115)
           (* (cos re) (- im))
           (if (<= im 3.8e+241) t_1 (if (<= im 7.2e+280) t_3 t_2))))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double t_1 = 0.5 * (exp(-im) - exp(im));
	double t_2 = t_0 - im;
	double t_3 = t_2 * ((-0.5 * (re * re)) + 1.0);
	double tmp;
	if (im <= -2e+238) {
		tmp = t_0;
	} else if (im <= -2e+167) {
		tmp = t_3;
	} else if (im <= -0.00098) {
		tmp = t_1;
	} else if (im <= 0.00115) {
		tmp = cos(re) * -im;
	} else if (im <= 3.8e+241) {
		tmp = t_1;
	} else if (im <= 7.2e+280) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_1 = 0.5d0 * (exp(-im) - exp(im))
    t_2 = t_0 - im
    t_3 = t_2 * (((-0.5d0) * (re * re)) + 1.0d0)
    if (im <= (-2d+238)) then
        tmp = t_0
    else if (im <= (-2d+167)) then
        tmp = t_3
    else if (im <= (-0.00098d0)) then
        tmp = t_1
    else if (im <= 0.00115d0) then
        tmp = cos(re) * -im
    else if (im <= 3.8d+241) then
        tmp = t_1
    else if (im <= 7.2d+280) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double t_1 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_2 = t_0 - im;
	double t_3 = t_2 * ((-0.5 * (re * re)) + 1.0);
	double tmp;
	if (im <= -2e+238) {
		tmp = t_0;
	} else if (im <= -2e+167) {
		tmp = t_3;
	} else if (im <= -0.00098) {
		tmp = t_1;
	} else if (im <= 0.00115) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 3.8e+241) {
		tmp = t_1;
	} else if (im <= 7.2e+280) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	t_1 = 0.5 * (math.exp(-im) - math.exp(im))
	t_2 = t_0 - im
	t_3 = t_2 * ((-0.5 * (re * re)) + 1.0)
	tmp = 0
	if im <= -2e+238:
		tmp = t_0
	elif im <= -2e+167:
		tmp = t_3
	elif im <= -0.00098:
		tmp = t_1
	elif im <= 0.00115:
		tmp = math.cos(re) * -im
	elif im <= 3.8e+241:
		tmp = t_1
	elif im <= 7.2e+280:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_1 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_2 = Float64(t_0 - im)
	t_3 = Float64(t_2 * Float64(Float64(-0.5 * Float64(re * re)) + 1.0))
	tmp = 0.0
	if (im <= -2e+238)
		tmp = t_0;
	elseif (im <= -2e+167)
		tmp = t_3;
	elseif (im <= -0.00098)
		tmp = t_1;
	elseif (im <= 0.00115)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 3.8e+241)
		tmp = t_1;
	elseif (im <= 7.2e+280)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	t_1 = 0.5 * (exp(-im) - exp(im));
	t_2 = t_0 - im;
	t_3 = t_2 * ((-0.5 * (re * re)) + 1.0);
	tmp = 0.0;
	if (im <= -2e+238)
		tmp = t_0;
	elseif (im <= -2e+167)
		tmp = t_3;
	elseif (im <= -0.00098)
		tmp = t_1;
	elseif (im <= 0.00115)
		tmp = cos(re) * -im;
	elseif (im <= 3.8e+241)
		tmp = t_1;
	elseif (im <= 7.2e+280)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - im), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2e+238], t$95$0, If[LessEqual[im, -2e+167], t$95$3, If[LessEqual[im, -0.00098], t$95$1, If[LessEqual[im, 0.00115], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 3.8e+241], t$95$1, If[LessEqual[im, 7.2e+280], t$95$3, t$95$2]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -2.0000000000000001e238

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 92.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   8. Taylor expanded in im around inf 92.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]

   if -2.0000000000000001e238 < im < -2.0000000000000001e167 or 3.79999999999999972e241 < im < 7.1999999999999999e280

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \color{blue}{{im}^{3} \cdot -0.16666666666666666}\right) - im \]

      2. associate-+r- 0.0%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

      3. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      4. *-commutative 0.0%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      5. fma-neg 0.0%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, -im\right)} + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      6. fma-neg 0.0%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, -im\right) + \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, -im\right)} \]

      7. *-lft-identity 0.0%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, -im\right) + \color{blue}{1 \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, -im\right)} \]

      8. distribute-rgt-out 82.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, -im\right) \cdot \left(-0.5 \cdot {re}^{2} + 1\right)} \]

      9. fma-neg 82.1%

         \[\leadsto \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 - im\right)} \cdot \left(-0.5 \cdot {re}^{2} + 1\right) \]

      10. *-commutative 82.1%

          \[\leadsto \left(\color{blue}{-0.16666666666666666 \cdot {im}^{3}} - im\right) \cdot \left(-0.5 \cdot {re}^{2} + 1\right) \]

      11. unpow2 82.1%

          \[\leadsto \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \]

   9. Simplified 82.1%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)} \]

   if -2.0000000000000001e167 < im < -9.7999999999999997e-4 or 0.00115 < im < 3.79999999999999972e241

   1. Initial program 99.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in re around 0 83.6%

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

   if -9.7999999999999997e-4 < im < 0.00115

   1. Initial program 7.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 7.7%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in im around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. distribute-lft-neg-in 99.6%

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

   6. Simplified 99.6%

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

   if 7.1999999999999999e280 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 71.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{+238}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -2 \cdot 10^{+167}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{elif}\;im \leq -0.00098:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.00115:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+241}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+280}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 5: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 12:\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984 - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 12.0)
   (- (* (pow im 7.0) -0.0001984126984126984) im)
   (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 12.0) {
		tmp = (pow(im, 7.0) * -0.0001984126984126984) - im;
	} else {
		tmp = cos(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 (re <= 12.0d0) then
        tmp = ((im ** 7.0d0) * (-0.0001984126984126984d0)) - im
    else
        tmp = cos(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 (re <= 12.0) {
		tmp = (Math.pow(im, 7.0) * -0.0001984126984126984) - im;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 12.0:
		tmp = (math.pow(im, 7.0) * -0.0001984126984126984) - im
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 12.0)
		tmp = Float64(Float64((im ^ 7.0) * -0.0001984126984126984) - im);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 12.0)
		tmp = ((im ^ 7.0) * -0.0001984126984126984) - im;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 12.0], N[(N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] - im), $MachinePrecision], N[(N[Cos[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 re < 12

   1. Initial program 51.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in im around 0 94.8%

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

      1. +-commutative 94.8%

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

      2. associate-+r+ 94.8%

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

      3. associate-+l+ 94.8%

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

      4. mul-1-neg 94.8%

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

      5. distribute-rgt-neg-in 94.8%

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

      6. *-commutative 94.8%

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

      7. associate-*l* 94.8%

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

      8. distribute-lft-out 94.8%

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

      9. *-commutative 94.8%

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

      10. associate-*r* 94.8%

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

      11. *-commutative 94.8%

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

      12. associate-*r* 94.8%

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

   6. Simplified 94.8%

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

   7. Taylor expanded in re around 0 72.1%

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

   8. Taylor expanded in im around inf 71.5%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot {im}^{7}} - im \]

   if 12 < re

   1. Initial program 43.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in im around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. *-commutative 98.2%

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

      4. associate-*l* 98.2%

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

      5. distribute-lft-out-- 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 12:\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984 - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 6: 81.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ t_1 := {im}^{7} \cdot -0.0001984126984126984 - im\\ t_2 := t_0 - im\\ t_3 := t_2 \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{if}\;im \leq -5 \cdot 10^{+237}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6 \cdot 10^{+165}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq -1.1 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 15000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+280}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666))
        (t_1 (- (* (pow im 7.0) -0.0001984126984126984) im))
        (t_2 (- t_0 im))
        (t_3 (* t_2 (+ (* -0.5 (* re re)) 1.0))))
   (if (<= im -5e+237)
     t_0
     (if (<= im -6e+165)
       t_3
       (if (<= im -1.1e+37)
         t_1
         (if (<= im 15000.0)
           (* (cos re) (- im))
           (if (<= im 3.5e+241) t_1 (if (<= im 8.2e+280) t_3 t_2))))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (pow(im, 7.0) * -0.0001984126984126984) - im;
	double t_2 = t_0 - im;
	double t_3 = t_2 * ((-0.5 * (re * re)) + 1.0);
	double tmp;
	if (im <= -5e+237) {
		tmp = t_0;
	} else if (im <= -6e+165) {
		tmp = t_3;
	} else if (im <= -1.1e+37) {
		tmp = t_1;
	} else if (im <= 15000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 3.5e+241) {
		tmp = t_1;
	} else if (im <= 8.2e+280) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (im ** 3.0d0) * (-0.16666666666666666d0)
    t_1 = ((im ** 7.0d0) * (-0.0001984126984126984d0)) - im
    t_2 = t_0 - im
    t_3 = t_2 * (((-0.5d0) * (re * re)) + 1.0d0)
    if (im <= (-5d+237)) then
        tmp = t_0
    else if (im <= (-6d+165)) then
        tmp = t_3
    else if (im <= (-1.1d+37)) then
        tmp = t_1
    else if (im <= 15000.0d0) then
        tmp = cos(re) * -im
    else if (im <= 3.5d+241) then
        tmp = t_1
    else if (im <= 8.2d+280) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 3.0) * -0.16666666666666666;
	double t_1 = (Math.pow(im, 7.0) * -0.0001984126984126984) - im;
	double t_2 = t_0 - im;
	double t_3 = t_2 * ((-0.5 * (re * re)) + 1.0);
	double tmp;
	if (im <= -5e+237) {
		tmp = t_0;
	} else if (im <= -6e+165) {
		tmp = t_3;
	} else if (im <= -1.1e+37) {
		tmp = t_1;
	} else if (im <= 15000.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 3.5e+241) {
		tmp = t_1;
	} else if (im <= 8.2e+280) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	t_1 = (math.pow(im, 7.0) * -0.0001984126984126984) - im
	t_2 = t_0 - im
	t_3 = t_2 * ((-0.5 * (re * re)) + 1.0)
	tmp = 0
	if im <= -5e+237:
		tmp = t_0
	elif im <= -6e+165:
		tmp = t_3
	elif im <= -1.1e+37:
		tmp = t_1
	elif im <= 15000.0:
		tmp = math.cos(re) * -im
	elif im <= 3.5e+241:
		tmp = t_1
	elif im <= 8.2e+280:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	t_1 = Float64(Float64((im ^ 7.0) * -0.0001984126984126984) - im)
	t_2 = Float64(t_0 - im)
	t_3 = Float64(t_2 * Float64(Float64(-0.5 * Float64(re * re)) + 1.0))
	tmp = 0.0
	if (im <= -5e+237)
		tmp = t_0;
	elseif (im <= -6e+165)
		tmp = t_3;
	elseif (im <= -1.1e+37)
		tmp = t_1;
	elseif (im <= 15000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 3.5e+241)
		tmp = t_1;
	elseif (im <= 8.2e+280)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	t_1 = ((im ^ 7.0) * -0.0001984126984126984) - im;
	t_2 = t_0 - im;
	t_3 = t_2 * ((-0.5 * (re * re)) + 1.0);
	tmp = 0.0;
	if (im <= -5e+237)
		tmp = t_0;
	elseif (im <= -6e+165)
		tmp = t_3;
	elseif (im <= -1.1e+37)
		tmp = t_1;
	elseif (im <= 15000.0)
		tmp = cos(re) * -im;
	elseif (im <= 3.5e+241)
		tmp = t_1;
	elseif (im <= 8.2e+280)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] - im), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - im), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5e+237], t$95$0, If[LessEqual[im, -6e+165], t$95$3, If[LessEqual[im, -1.1e+37], t$95$1, If[LessEqual[im, 15000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 3.5e+241], t$95$1, If[LessEqual[im, 8.2e+280], t$95$3, t$95$2]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -5.0000000000000002e237

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 92.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   8. Taylor expanded in im around inf 92.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]

   if -5.0000000000000002e237 < im < -5.99999999999999981e165 or 3.5e241 < im < 8.19999999999999995e280

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \color{blue}{{im}^{3} \cdot -0.16666666666666666}\right) - im \]

      2. associate-+r- 0.0%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

      3. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      4. *-commutative 0.0%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      5. fma-neg 0.0%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, -im\right)} + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      6. fma-neg 0.0%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, -im\right) + \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, -im\right)} \]

      7. *-lft-identity 0.0%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, -im\right) + \color{blue}{1 \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, -im\right)} \]

      8. distribute-rgt-out 82.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, -im\right) \cdot \left(-0.5 \cdot {re}^{2} + 1\right)} \]

      9. fma-neg 82.1%

         \[\leadsto \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 - im\right)} \cdot \left(-0.5 \cdot {re}^{2} + 1\right) \]

      10. *-commutative 82.1%

          \[\leadsto \left(\color{blue}{-0.16666666666666666 \cdot {im}^{3}} - im\right) \cdot \left(-0.5 \cdot {re}^{2} + 1\right) \]

      11. unpow2 82.1%

          \[\leadsto \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \]

   9. Simplified 82.1%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)} \]

   if -5.99999999999999981e165 < im < -1.1e37 or 15000 < im < 3.5e241

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 90.9%

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

      1. +-commutative 90.9%

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

      2. associate-+r+ 90.9%

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

      3. associate-+l+ 90.9%

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

      4. mul-1-neg 90.9%

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

      5. distribute-rgt-neg-in 90.9%

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

      6. *-commutative 90.9%

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

      7. associate-*l* 90.9%

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

      8. distribute-lft-out 90.9%

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

      9. *-commutative 90.9%

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

      10. associate-*r* 90.9%

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

      11. *-commutative 90.9%

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

      12. associate-*r* 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in re around 0 76.2%

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

   8. Taylor expanded in im around inf 76.2%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot {im}^{7}} - im \]

   if -1.1e37 < im < 15000

   1. Initial program 12.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 12.0%

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

   3. Simplified 12.0%

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

   4. Taylor expanded in im around 0 95.6%

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

      1. mul-1-neg 95.6%

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

      2. *-commutative 95.6%

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

      3. distribute-lft-neg-in 95.6%

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

   6. Simplified 95.6%

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

   if 8.19999999999999995e280 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 71.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+237}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -6 \cdot 10^{+165}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{elif}\;im \leq -1.1 \cdot 10^{+37}:\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984 - im\\ \mathbf{elif}\;im \leq 15000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+241}:\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984 - im\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+280}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 7: 75.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.00044 \lor \neg \left(im \leq 0.00062\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.00044) (not (<= im 0.00062)))
   (- (* (pow im 3.0) -0.16666666666666666) im)
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.00044) || !(im <= 0.00062)) {
		tmp = (pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = cos(re) * -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 <= (-0.00044d0)) .or. (.not. (im <= 0.00062d0))) then
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.00044) || !(im <= 0.00062)) {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.00044) or not (im <= 0.00062):
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.00044) || !(im <= 0.00062))
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.00044) || ~((im <= 0.00062)))
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.00044], N[Not[LessEqual[im, 0.00062]], $MachinePrecision]], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.40000000000000016e-4 or 6.2e-4 < im

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in im around 0 82.5%

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

      1. mul-1-neg 82.5%

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

      2. unsub-neg 82.5%

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

      3. *-commutative 82.5%

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

      4. associate-*l* 82.5%

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

      5. distribute-lft-out-- 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in re around 0 60.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   if -4.40000000000000016e-4 < im < 6.2e-4

   1. Initial program 7.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 7.7%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in im around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. distribute-lft-neg-in 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.00044 \lor \neg \left(im \leq 0.00062\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 8: 81.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+37} \lor \neg \left(im \leq 7.1\right):\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984 - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.1e+37) (not (<= im 7.1)))
   (- (* (pow im 7.0) -0.0001984126984126984) im)
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.1e+37) || !(im <= 7.1)) {
		tmp = (pow(im, 7.0) * -0.0001984126984126984) - im;
	} else {
		tmp = cos(re) * -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 <= (-1.1d+37)) .or. (.not. (im <= 7.1d0))) then
        tmp = ((im ** 7.0d0) * (-0.0001984126984126984d0)) - im
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.1e+37) || !(im <= 7.1)) {
		tmp = (Math.pow(im, 7.0) * -0.0001984126984126984) - im;
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.1e+37) or not (im <= 7.1):
		tmp = (math.pow(im, 7.0) * -0.0001984126984126984) - im
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.1e+37) || !(im <= 7.1))
		tmp = Float64(Float64((im ^ 7.0) * -0.0001984126984126984) - im);
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.1e+37) || ~((im <= 7.1)))
		tmp = ((im ^ 7.0) * -0.0001984126984126984) - im;
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.1e+37], N[Not[LessEqual[im, 7.1]], $MachinePrecision]], N[(N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] - im), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.1e37 or 7.0999999999999996 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 94.9%

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

      1. +-commutative 94.9%

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

      2. associate-+r+ 94.9%

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

      3. associate-+l+ 94.9%

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

      4. mul-1-neg 94.9%

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

      5. distribute-rgt-neg-in 94.9%

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

      6. *-commutative 94.9%

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

      7. associate-*l* 94.9%

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

      8. distribute-lft-out 94.9%

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

      9. *-commutative 94.9%

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

      10. associate-*r* 94.9%

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

      11. *-commutative 94.9%

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

      12. associate-*r* 94.9%

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

   6. Simplified 94.9%

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

   7. Taylor expanded in re around 0 70.2%

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

   8. Taylor expanded in im around inf 70.2%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot {im}^{7}} - im \]

   if -1.1e37 < im < 7.0999999999999996

   1. Initial program 12.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 12.0%

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

   3. Simplified 12.0%

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

   4. Taylor expanded in im around 0 95.6%

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

      1. mul-1-neg 95.6%

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

      2. *-commutative 95.6%

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

      3. distribute-lft-neg-in 95.6%

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

   6. Simplified 95.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+37} \lor \neg \left(im \leq 7.1\right):\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984 - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 9: 53.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.5 \lor \neg \left(im \leq 2.5\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.5) (not (<= im 2.5)))
   (* (pow im 3.0) -0.16666666666666666)
   (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.5) || !(im <= 2.5)) {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	} else {
		tmp = -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 <= (-2.5d0)) .or. (.not. (im <= 2.5d0))) then
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.5) || !(im <= 2.5)) {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.5) or not (im <= 2.5):
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.5) || !(im <= 2.5))
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.5) || ~((im <= 2.5)))
		tmp = (im ^ 3.0) * -0.16666666666666666;
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.5], N[Not[LessEqual[im, 2.5]], $MachinePrecision]], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -2.5 or 2.5 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 82.5%

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

      1. mul-1-neg 82.5%

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

      2. unsub-neg 82.5%

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

      3. *-commutative 82.5%

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

      4. associate-*l* 82.5%

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

      5. distribute-lft-out-- 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in re around 0 59.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   8. Taylor expanded in im around inf 59.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]

   if -2.5 < im < 2.5

   1. Initial program 8.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 8.9%

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

   3. Simplified 8.9%

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

   4. Taylor expanded in im around 0 98.9%

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

      1. mul-1-neg 98.9%

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

      2. *-commutative 98.9%

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

      3. distribute-lft-neg-in 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in re around 0 51.6%

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

      1. neg-mul-1 51.6%

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

      2. +-commutative 51.6%

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

      3. unsub-neg 51.6%

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

      4. *-commutative 51.6%

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

      5. *-commutative 51.6%

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

      6. associate-*l* 51.6%

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

      7. unpow2 51.6%

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

   9. Simplified 51.6%

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

   10. Taylor expanded in re around 0 55.6%

       \[\leadsto \color{blue}{-1 \cdot im} \]
   11. Step-by-step derivation

       1. neg-mul-1 55.6%

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

   12. Simplified 55.6%

       \[\leadsto \color{blue}{-im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.5 \lor \neg \left(im \leq 2.5\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 10: 75.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.15 \cdot 10^{+37} \lor \neg \left(im \leq 15000\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.15e+37) (not (<= im 15000.0)))
   (* (pow im 3.0) -0.16666666666666666)
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.15e+37) || !(im <= 15000.0)) {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	} else {
		tmp = cos(re) * -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 <= (-1.15d+37)) .or. (.not. (im <= 15000.0d0))) then
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.15e+37) || !(im <= 15000.0)) {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.15e+37) or not (im <= 15000.0):
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.15e+37) || !(im <= 15000.0))
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.15e+37) || ~((im <= 15000.0)))
		tmp = (im ^ 3.0) * -0.16666666666666666;
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.15e+37], N[Not[LessEqual[im, 15000.0]], $MachinePrecision]], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.15000000000000001e37 or 15000 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 86.2%

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

      1. mul-1-neg 86.2%

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

      2. unsub-neg 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-*l* 86.2%

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

      5. distribute-lft-out-- 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in re around 0 62.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   8. Taylor expanded in im around inf 62.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]

   if -1.15000000000000001e37 < im < 15000

   1. Initial program 12.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 12.0%

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

   3. Simplified 12.0%

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

   4. Taylor expanded in im around 0 95.6%

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

      1. mul-1-neg 95.6%

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

      2. *-commutative 95.6%

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

      3. distribute-lft-neg-in 95.6%

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

   6. Simplified 95.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.15 \cdot 10^{+37} \lor \neg \left(im \leq 15000\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 11: 35.6% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + re \cdot \left(re \cdot -0.25\right)\\ \mathbf{if}\;re \leq 1.95 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+257} \lor \neg \left(re \leq 4.3 \cdot 10^{+287}\right):\\ \;\;\;\;-3 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 27\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* re (* re -0.25)))))
   (if (<= re 1.95e+154)
     (- (* im (* 0.5 (* re re))) im)
     (if (or (<= re 5.5e+257) (not (<= re 4.3e+287)))
       (* -3.0 t_0)
       (* t_0 27.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 + (re * (re * -0.25));
	double tmp;
	if (re <= 1.95e+154) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else if ((re <= 5.5e+257) || !(re <= 4.3e+287)) {
		tmp = -3.0 * t_0;
	} else {
		tmp = t_0 * 27.0;
	}
	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 = 0.5d0 + (re * (re * (-0.25d0)))
    if (re <= 1.95d+154) then
        tmp = (im * (0.5d0 * (re * re))) - im
    else if ((re <= 5.5d+257) .or. (.not. (re <= 4.3d+287))) then
        tmp = (-3.0d0) * t_0
    else
        tmp = t_0 * 27.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + (re * (re * -0.25));
	double tmp;
	if (re <= 1.95e+154) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else if ((re <= 5.5e+257) || !(re <= 4.3e+287)) {
		tmp = -3.0 * t_0;
	} else {
		tmp = t_0 * 27.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + (re * (re * -0.25))
	tmp = 0
	if re <= 1.95e+154:
		tmp = (im * (0.5 * (re * re))) - im
	elif (re <= 5.5e+257) or not (re <= 4.3e+287):
		tmp = -3.0 * t_0
	else:
		tmp = t_0 * 27.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(re * Float64(re * -0.25)))
	tmp = 0.0
	if (re <= 1.95e+154)
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	elseif ((re <= 5.5e+257) || !(re <= 4.3e+287))
		tmp = Float64(-3.0 * t_0);
	else
		tmp = Float64(t_0 * 27.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + (re * (re * -0.25));
	tmp = 0.0;
	if (re <= 1.95e+154)
		tmp = (im * (0.5 * (re * re))) - im;
	elseif ((re <= 5.5e+257) || ~((re <= 4.3e+287)))
		tmp = -3.0 * t_0;
	else
		tmp = t_0 * 27.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 1.95e+154], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[Or[LessEqual[re, 5.5e+257], N[Not[LessEqual[re, 4.3e+287]], $MachinePrecision]], N[(-3.0 * t$95$0), $MachinePrecision], N[(t$95$0 * 27.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.9500000000000001e154

   1. Initial program 49.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 49.9%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in im around 0 57.0%

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

      1. mul-1-neg 57.0%

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

      2. *-commutative 57.0%

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

      3. distribute-lft-neg-in 57.0%

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

   6. Simplified 57.0%

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

   7. Taylor expanded in re around 0 39.1%

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

      1. neg-mul-1 39.1%

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

      2. +-commutative 39.1%

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

      3. unsub-neg 39.1%

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

      4. *-commutative 39.1%

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

      5. *-commutative 39.1%

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

      6. associate-*l* 39.1%

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

      7. unpow2 39.1%

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

   9. Simplified 39.1%

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

   if 1.9500000000000001e154 < re < 5.49999999999999957e257 or 4.3e287 < re

   1. Initial program 41.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 41.2%

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

   3. Simplified 41.2%

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

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 25.0%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 25.0%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 25.0%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 25.0%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 25.0%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 25.0%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 32.0%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   if 5.49999999999999957e257 < re < 4.3e287

   1. Initial program 54.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 54.7%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in re around 0 0.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.1%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.1%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 28.7%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 28.7%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 28.7%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 28.7%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 28.7%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 28.7%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.95 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{+257} \lor \neg \left(re \leq 4.3 \cdot 10^{+287}\right):\\ \;\;\;\;-3 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \end{array} \]

Alternative 12: 32.4% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + re \cdot \left(re \cdot -0.25\right)\\ \mathbf{if}\;im \leq -1.02 \cdot 10^{+42}:\\ \;\;\;\;-3 \cdot t_0\\ \mathbf{elif}\;im \leq 4.9 \cdot 10^{+179}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 27\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* re (* re -0.25)))))
   (if (<= im -1.02e+42)
     (* -3.0 t_0)
     (if (<= im 4.9e+179) (- im) (* t_0 27.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 + (re * (re * -0.25));
	double tmp;
	if (im <= -1.02e+42) {
		tmp = -3.0 * t_0;
	} else if (im <= 4.9e+179) {
		tmp = -im;
	} else {
		tmp = t_0 * 27.0;
	}
	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 = 0.5d0 + (re * (re * (-0.25d0)))
    if (im <= (-1.02d+42)) then
        tmp = (-3.0d0) * t_0
    else if (im <= 4.9d+179) then
        tmp = -im
    else
        tmp = t_0 * 27.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 + (re * (re * -0.25));
	double tmp;
	if (im <= -1.02e+42) {
		tmp = -3.0 * t_0;
	} else if (im <= 4.9e+179) {
		tmp = -im;
	} else {
		tmp = t_0 * 27.0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 + (re * (re * -0.25))
	tmp = 0
	if im <= -1.02e+42:
		tmp = -3.0 * t_0
	elif im <= 4.9e+179:
		tmp = -im
	else:
		tmp = t_0 * 27.0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 + Float64(re * Float64(re * -0.25)))
	tmp = 0.0
	if (im <= -1.02e+42)
		tmp = Float64(-3.0 * t_0);
	elseif (im <= 4.9e+179)
		tmp = Float64(-im);
	else
		tmp = Float64(t_0 * 27.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 + (re * (re * -0.25));
	tmp = 0.0;
	if (im <= -1.02e+42)
		tmp = -3.0 * t_0;
	elseif (im <= 4.9e+179)
		tmp = -im;
	else
		tmp = t_0 * 27.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.02e+42], N[(-3.0 * t$95$0), $MachinePrecision], If[LessEqual[im, 4.9e+179], (-im), N[(t$95$0 * 27.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.01999999999999996e42

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 63.3%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 63.3%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 63.3%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 63.3%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 63.3%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 63.3%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   if -1.01999999999999996e42 < im < 4.8999999999999999e179

   1. Initial program 23.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 23.4%

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

   3. Simplified 23.4%

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

   4. Taylor expanded in im around 0 83.7%

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

      1. mul-1-neg 83.7%

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

      2. *-commutative 83.7%

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

      3. distribute-lft-neg-in 83.7%

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

   6. Simplified 83.7%

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

   7. Taylor expanded in re around 0 46.2%

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

      1. neg-mul-1 46.2%

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

      2. +-commutative 46.2%

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

      3. unsub-neg 46.2%

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

      4. *-commutative 46.2%

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

      5. *-commutative 46.2%

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

      6. associate-*l* 46.2%

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

      7. unpow2 46.2%

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

   9. Simplified 46.2%

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

   10. Taylor expanded in re around 0 47.2%

       \[\leadsto \color{blue}{-1 \cdot im} \]
   11. Step-by-step derivation

       1. neg-mul-1 47.2%

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

   12. Simplified 47.2%

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

   if 4.8999999999999999e179 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 55.3%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 55.3%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 55.3%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 55.3%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 55.3%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 55.3%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 30.2%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.02 \cdot 10^{+42}:\\ \;\;\;\;-3 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 4.9 \cdot 10^{+179}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \end{array} \]

Alternative 13: 31.8% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{+153}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.9e+153) (- im) (* -3.0 (+ 0.5 (* re (* re -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.9e+153) {
		tmp = -im;
	} else {
		tmp = -3.0 * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.9d+153) then
        tmp = -im
    else
        tmp = (-3.0d0) * (0.5d0 + (re * (re * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.9e+153) {
		tmp = -im;
	} else {
		tmp = -3.0 * (0.5 + (re * (re * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.9e+153:
		tmp = -im
	else:
		tmp = -3.0 * (0.5 + (re * (re * -0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.9e+153)
		tmp = Float64(-im);
	else
		tmp = Float64(-3.0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.9e+153)
		tmp = -im;
	else
		tmp = -3.0 * (0.5 + (re * (re * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.9e+153], (-im), N[(-3.0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.89999999999999983e153

   1. Initial program 49.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 49.9%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in im around 0 57.0%

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

      1. mul-1-neg 57.0%

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

      2. *-commutative 57.0%

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

      3. distribute-lft-neg-in 57.0%

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

   6. Simplified 57.0%

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

   7. Taylor expanded in re around 0 39.1%

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

      1. neg-mul-1 39.1%

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

      2. +-commutative 39.1%

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

      3. unsub-neg 39.1%

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

      4. *-commutative 39.1%

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

      5. *-commutative 39.1%

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

      6. associate-*l* 39.1%

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

      7. unpow2 39.1%

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

   9. Simplified 39.1%

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

   10. Taylor expanded in re around 0 35.3%

       \[\leadsto \color{blue}{-1 \cdot im} \]
   11. Step-by-step derivation

       1. neg-mul-1 35.3%

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

   12. Simplified 35.3%

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

   if 1.89999999999999983e153 < re

   1. Initial program 45.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 26.1%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 26.1%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 26.1%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 26.1%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 26.1%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 26.1%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 27.0%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{+153}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 14: 29.3% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 49.5%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in im around 0 57.5%

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

   1. mul-1-neg 57.5%

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

   2. *-commutative 57.5%

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

   3. distribute-lft-neg-in 57.5%

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

6. Simplified 57.5%

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

7. Taylor expanded in re around 0 38.0%

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

   1. neg-mul-1 38.0%

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

   2. +-commutative 38.0%

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

   3. unsub-neg 38.0%

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

   4. *-commutative 38.0%

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

   5. *-commutative 38.0%

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

   6. associate-*l* 38.0%

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

   7. unpow2 38.0%

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

9. Simplified 38.0%

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

10. Taylor expanded in re around 0 32.8%

    \[\leadsto \color{blue}{-1 \cdot im} \]
11. Step-by-step derivation

    1. neg-mul-1 32.8%

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

12. Simplified 32.8%

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

13. Final simplification 32.8%

    \[\leadsto -im \]

Developer target: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - 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 = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - 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.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[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[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023238 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))