math.sin on complex, imaginary part

Percentage Accurate: 54.7% → 98.9%

Time: 9.7s

Alternatives: 11

Speedup: 2.8×

• 49.7% 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: 54.7% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-2 \cdot \cos re\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (log1p (expm1 (* im (* -2.0 (cos re)))))))

C

double code(double re, double im) {
	return 0.5 * log1p(expm1((im * (-2.0 * cos(re)))));
}

Java

public static double code(double re, double im) {
	return 0.5 * Math.log1p(Math.expm1((im * (-2.0 * Math.cos(re)))));
}

Python

def code(re, im):
	return 0.5 * math.log1p(math.expm1((im * (-2.0 * math.cos(re)))))

Julia

function code(re, im)
	return Float64(0.5 * log1p(expm1(Float64(im * Float64(-2.0 * cos(re))))))
end

Wolfram

code[re_, im_] := N[(0.5 * N[Log[1 + N[(Exp[N[(im * N[(-2.0 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

   1. /-rgt-identity 50.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 50.9%

      \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. associate-*l/ 50.9%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   4. cos-neg 50.9%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

   5. associate-*l* 50.9%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

   6. associate-*r/ 50.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   7. exp-0 50.9%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 50.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   9. *-commutative 50.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

   10. neg-sub0 50.9%

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

   11. cos-neg 50.9%

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

3. Simplified 50.9%

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

5. Taylor expanded in im around 0 55.2%

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

   1. log1p-expm1-u 98.8%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot im\right) \cdot \cos re\right)\right)} \]

   2. *-commutative 98.8%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(im \cdot -2\right)} \cdot \cos re\right)\right) \]

   3. associate-*l* 98.8%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{im \cdot \left(-2 \cdot \cos re\right)}\right)\right) \]

7. Applied egg-rr 98.8%

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-2 \cdot \cos re\right)\right)\right)} \]
8. Add Preprocessing

Alternative 2: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 380:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333\right) + im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 380.0)
   (*
    0.5
    (* (cos re) (+ (* im (* (pow im 2.0) -0.3333333333333333)) (* im -2.0))))
   (if (<= im 1.06e+44)
     (* 0.5 (log1p (expm1 (* -0.0003968253968253968 (pow im 7.0)))))
     (* 0.5 (* -0.0003968253968253968 (* (cos re) (pow im 7.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 380.0) {
		tmp = 0.5 * (cos(re) * ((im * (pow(im, 2.0) * -0.3333333333333333)) + (im * -2.0)));
	} else if (im <= 1.06e+44) {
		tmp = 0.5 * log1p(expm1((-0.0003968253968253968 * pow(im, 7.0))));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * (cos(re) * pow(im, 7.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 380.0) {
		tmp = 0.5 * (Math.cos(re) * ((im * (Math.pow(im, 2.0) * -0.3333333333333333)) + (im * -2.0)));
	} else if (im <= 1.06e+44) {
		tmp = 0.5 * Math.log1p(Math.expm1((-0.0003968253968253968 * Math.pow(im, 7.0))));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * (Math.cos(re) * Math.pow(im, 7.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 380.0:
		tmp = 0.5 * (math.cos(re) * ((im * (math.pow(im, 2.0) * -0.3333333333333333)) + (im * -2.0)))
	elif im <= 1.06e+44:
		tmp = 0.5 * math.log1p(math.expm1((-0.0003968253968253968 * math.pow(im, 7.0))))
	else:
		tmp = 0.5 * (-0.0003968253968253968 * (math.cos(re) * math.pow(im, 7.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 380.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(im * Float64((im ^ 2.0) * -0.3333333333333333)) + Float64(im * -2.0))));
	elseif (im <= 1.06e+44)
		tmp = Float64(0.5 * log1p(expm1(Float64(-0.0003968253968253968 * (im ^ 7.0)))));
	else
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * Float64(cos(re) * (im ^ 7.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 380.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(im * N[(N[Power[im, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.06e+44], N[(0.5 * N[Log[1 + N[(Exp[N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.0003968253968253968 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 380

   1. Initial program 34.8%

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

      1. /-rgt-identity 34.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 34.8%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 34.8%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 34.8%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 34.8%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 34.8%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 34.8%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 34.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 34.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 34.8%

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

      11. cos-neg 34.8%

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

   3. Simplified 34.8%

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

   5. Taylor expanded in im around 0 89.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \cdot \cos re\right) \]
   6. Step-by-step derivation

      1. sub-neg 89.3%

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

      2. metadata-eval 89.3%

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

      3. distribute-rgt-in 89.3%

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

      4. *-commutative 89.3%

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

      5. *-commutative 89.3%

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

   7. Applied egg-rr 89.3%

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

   if 380 < im < 1.06e44

   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. /-rgt-identity 99.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 99.8%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 99.8%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 99.8%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 99.8%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 99.8%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 99.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 99.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 99.8%

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

      11. cos-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in im around 0 5.5%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 5.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 5.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 72.2%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)} \]

   9. Applied egg-rr 72.2%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)} \]

   if 1.06e44 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 380:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333\right) + im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1000000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1000000000000.0)
   (* 0.5 (* (cos re) (* im (- (* -0.3333333333333333 (* im im)) 2.0))))
   (if (<= im 1.06e+44)
     (* 0.5 (+ (* im -2.0) (* im (* -0.08333333333333333 (pow re 4.0)))))
     (* 0.5 (* -0.0003968253968253968 (* (cos re) (pow im 7.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1000000000000.0) {
		tmp = 0.5 * (cos(re) * (im * ((-0.3333333333333333 * (im * im)) - 2.0)));
	} else if (im <= 1.06e+44) {
		tmp = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * pow(re, 4.0))));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * (cos(re) * pow(im, 7.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1000000000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im * (((-0.3333333333333333d0) * (im * im)) - 2.0d0)))
    else if (im <= 1.06d+44) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + (im * ((-0.08333333333333333d0) * (re ** 4.0d0))))
    else
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (cos(re) * (im ** 7.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1000000000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im * ((-0.3333333333333333 * (im * im)) - 2.0)));
	} else if (im <= 1.06e+44) {
		tmp = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * Math.pow(re, 4.0))));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * (Math.cos(re) * Math.pow(im, 7.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1000000000000.0:
		tmp = 0.5 * (math.cos(re) * (im * ((-0.3333333333333333 * (im * im)) - 2.0)))
	elif im <= 1.06e+44:
		tmp = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * math.pow(re, 4.0))))
	else:
		tmp = 0.5 * (-0.0003968253968253968 * (math.cos(re) * math.pow(im, 7.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1000000000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0))));
	elseif (im <= 1.06e+44)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(im * Float64(-0.08333333333333333 * (re ^ 4.0)))));
	else
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * Float64(cos(re) * (im ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1000000000000.0)
		tmp = 0.5 * (cos(re) * (im * ((-0.3333333333333333 * (im * im)) - 2.0)));
	elseif (im <= 1.06e+44)
		tmp = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * (re ^ 4.0))));
	else
		tmp = 0.5 * (-0.0003968253968253968 * (cos(re) * (im ^ 7.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1000000000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.06e+44], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(im * N[(-0.08333333333333333 * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.0003968253968253968 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1e12

   1. Initial program 35.5%

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

      1. /-rgt-identity 35.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 35.5%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 35.5%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 35.5%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 35.5%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 35.5%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 35.5%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 35.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 35.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 35.5%

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

      11. cos-neg 35.5%

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

   3. Simplified 35.5%

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

   5. Taylor expanded in im around 0 88.4%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \cdot \cos re\right) \]
   6. Step-by-step derivation

      1. unpow2 88.4%

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

   7. Applied egg-rr 88.4%

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

   if 1e12 < im < 1.06e44

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

   6. Taylor expanded in re around 0 61.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right)} \]

   7. Taylor expanded in re around inf 61.3%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r* 61.3%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{4}}\right) \]

      2. *-commutative 61.3%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot {re}^{4}\right) \]

      3. associate-*l* 61.3%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)}\right) \]

   9. Simplified 61.3%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)}\right) \]

   if 1.06e44 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1000000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (* (cos re) (* im (- (* -0.0003968253968253968 (pow im 6.0)) 2.0)))))

C

double code(double re, double im) {
	return 0.5 * (cos(re) * (im * ((-0.0003968253968253968 * pow(im, 6.0)) - 2.0)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (cos(re) * (im * (((-0.0003968253968253968d0) * (im ** 6.0d0)) - 2.0d0)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (Math.cos(re) * (im * ((-0.0003968253968253968 * Math.pow(im, 6.0)) - 2.0)));
}

Python

def code(re, im):
	return 0.5 * (math.cos(re) * (im * ((-0.0003968253968253968 * math.pow(im, 6.0)) - 2.0)))

Julia

function code(re, im)
	return Float64(0.5 * Float64(cos(re) * Float64(im * Float64(Float64(-0.0003968253968253968 * (im ^ 6.0)) - 2.0))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (cos(re) * (im * ((-0.0003968253968253968 * (im ^ 6.0)) - 2.0)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * N[(N[(-0.0003968253968253968 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

   1. /-rgt-identity 50.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 50.9%

      \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. associate-*l/ 50.9%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   4. cos-neg 50.9%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

   5. associate-*l* 50.9%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

   6. associate-*r/ 50.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   7. exp-0 50.9%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 50.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   9. *-commutative 50.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

   10. neg-sub0 50.9%

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

   11. cos-neg 50.9%

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

3. Simplified 50.9%

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

5. Taylor expanded in im around 0 94.3%

   \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]

6. Taylor expanded in im around inf 94.1%

   \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(\color{blue}{-0.0003968253968253968 \cdot {im}^{6}} - 2\right)\right) \cdot \cos re\right) \]

7. Final simplification 94.1%

   \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)\right) \]
8. Add Preprocessing

Alternative 5: 65.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1000000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+260}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+294}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1000000000000.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 3.3e+44)
     (* 0.5 (+ (* im -2.0) (* im (* -0.08333333333333333 (pow re 4.0)))))
     (if (<= im 1.2e+260)
       (* 0.5 (* im (- (* -0.0003968253968253968 (pow im 6.0)) 2.0)))
       (if (<= im 6.5e+294)
         (* 0.5 (* im (+ -2.0 (pow re 2.0))))
         (* 0.5 (* -0.0003968253968253968 (pow im 7.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1000000000000.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 3.3e+44) {
		tmp = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * pow(re, 4.0))));
	} else if (im <= 1.2e+260) {
		tmp = 0.5 * (im * ((-0.0003968253968253968 * pow(im, 6.0)) - 2.0));
	} else if (im <= 6.5e+294) {
		tmp = 0.5 * (im * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * pow(im, 7.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1000000000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else if (im <= 3.3d+44) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + (im * ((-0.08333333333333333d0) * (re ** 4.0d0))))
    else if (im <= 1.2d+260) then
        tmp = 0.5d0 * (im * (((-0.0003968253968253968d0) * (im ** 6.0d0)) - 2.0d0))
    else if (im <= 6.5d+294) then
        tmp = 0.5d0 * (im * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (im ** 7.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1000000000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 3.3e+44) {
		tmp = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * Math.pow(re, 4.0))));
	} else if (im <= 1.2e+260) {
		tmp = 0.5 * (im * ((-0.0003968253968253968 * Math.pow(im, 6.0)) - 2.0));
	} else if (im <= 6.5e+294) {
		tmp = 0.5 * (im * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * Math.pow(im, 7.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1000000000000.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 3.3e+44:
		tmp = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * math.pow(re, 4.0))))
	elif im <= 1.2e+260:
		tmp = 0.5 * (im * ((-0.0003968253968253968 * math.pow(im, 6.0)) - 2.0))
	elif im <= 6.5e+294:
		tmp = 0.5 * (im * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = 0.5 * (-0.0003968253968253968 * math.pow(im, 7.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1000000000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 3.3e+44)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(im * Float64(-0.08333333333333333 * (re ^ 4.0)))));
	elseif (im <= 1.2e+260)
		tmp = Float64(0.5 * Float64(im * Float64(Float64(-0.0003968253968253968 * (im ^ 6.0)) - 2.0)));
	elseif (im <= 6.5e+294)
		tmp = Float64(0.5 * Float64(im * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * (im ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1000000000000.0)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	elseif (im <= 3.3e+44)
		tmp = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * (re ^ 4.0))));
	elseif (im <= 1.2e+260)
		tmp = 0.5 * (im * ((-0.0003968253968253968 * (im ^ 6.0)) - 2.0));
	elseif (im <= 6.5e+294)
		tmp = 0.5 * (im * (-2.0 + (re ^ 2.0)));
	else
		tmp = 0.5 * (-0.0003968253968253968 * (im ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1000000000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.3e+44], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(im * N[(-0.08333333333333333 * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+260], N[(0.5 * N[(im * N[(N[(-0.0003968253968253968 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.5e+294], N[(0.5 * N[(im * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 1e12

   1. Initial program 35.5%

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

      1. /-rgt-identity 35.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 35.5%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 35.5%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 35.5%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 35.5%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 35.5%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 35.5%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 35.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 35.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 35.5%

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

      11. cos-neg 35.5%

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

   3. Simplified 35.5%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 1e12 < im < 3.30000000000000013e44

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

   6. Taylor expanded in re around 0 61.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right)} \]

   7. Taylor expanded in re around inf 61.3%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r* 61.3%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{4}}\right) \]

      2. *-commutative 61.3%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot {re}^{4}\right) \]

      3. associate-*l* 61.3%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)}\right) \]

   9. Simplified 61.3%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)}\right) \]

   if 3.30000000000000013e44 < im < 1.20000000000000005e260

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(\color{blue}{-0.0003968253968253968 \cdot {im}^{6}} - 2\right)\right) \cdot \cos re\right) \]

   7. Taylor expanded in re around 0 76.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)} \]

   if 1.20000000000000005e260 < im < 6.49999999999999965e294

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 8.6%

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

   6. Taylor expanded in re around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. *-commutative 68.5%

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

      3. distribute-lft-out 68.5%

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

   8. Simplified 68.5%

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

   if 6.49999999999999965e294 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1000000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+260}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+294}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right)\right)\\ \mathbf{if}\;im \leq 1000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          0.5
          (* (cos re) (* im (- (* -0.3333333333333333 (* im im)) 2.0))))))
   (if (<= im 1000000000000.0)
     t_0
     (if (<= im 1.06e+44)
       (* 0.5 (+ (* im -2.0) (* im (* -0.08333333333333333 (pow re 4.0)))))
       (if (<= im 8.2e+102)
         (* 0.5 (* -0.0003968253968253968 (pow im 7.0)))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (cos(re) * (im * ((-0.3333333333333333 * (im * im)) - 2.0)));
	double tmp;
	if (im <= 1000000000000.0) {
		tmp = t_0;
	} else if (im <= 1.06e+44) {
		tmp = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * pow(re, 4.0))));
	} else if (im <= 8.2e+102) {
		tmp = 0.5 * (-0.0003968253968253968 * pow(im, 7.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (cos(re) * (im * (((-0.3333333333333333d0) * (im * im)) - 2.0d0)))
    if (im <= 1000000000000.0d0) then
        tmp = t_0
    else if (im <= 1.06d+44) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + (im * ((-0.08333333333333333d0) * (re ** 4.0d0))))
    else if (im <= 8.2d+102) then
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (im ** 7.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.cos(re) * (im * ((-0.3333333333333333 * (im * im)) - 2.0)));
	double tmp;
	if (im <= 1000000000000.0) {
		tmp = t_0;
	} else if (im <= 1.06e+44) {
		tmp = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * Math.pow(re, 4.0))));
	} else if (im <= 8.2e+102) {
		tmp = 0.5 * (-0.0003968253968253968 * Math.pow(im, 7.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.cos(re) * (im * ((-0.3333333333333333 * (im * im)) - 2.0)))
	tmp = 0
	if im <= 1000000000000.0:
		tmp = t_0
	elif im <= 1.06e+44:
		tmp = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * math.pow(re, 4.0))))
	elif im <= 8.2e+102:
		tmp = 0.5 * (-0.0003968253968253968 * math.pow(im, 7.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(cos(re) * Float64(im * Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0))))
	tmp = 0.0
	if (im <= 1000000000000.0)
		tmp = t_0;
	elseif (im <= 1.06e+44)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(im * Float64(-0.08333333333333333 * (re ^ 4.0)))));
	elseif (im <= 8.2e+102)
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * (im ^ 7.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (cos(re) * (im * ((-0.3333333333333333 * (im * im)) - 2.0)));
	tmp = 0.0;
	if (im <= 1000000000000.0)
		tmp = t_0;
	elseif (im <= 1.06e+44)
		tmp = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * (re ^ 4.0))));
	elseif (im <= 8.2e+102)
		tmp = 0.5 * (-0.0003968253968253968 * (im ^ 7.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1000000000000.0], t$95$0, If[LessEqual[im, 1.06e+44], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(im * N[(-0.08333333333333333 * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.2e+102], N[(0.5 * N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1e12 or 8.1999999999999999e102 < im

   1. Initial program 47.6%

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

      1. /-rgt-identity 47.6%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 47.6%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 47.6%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 47.6%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 47.6%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 47.6%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 47.6%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 47.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 47.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 47.6%

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

      11. cos-neg 47.6%

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

   3. Simplified 47.6%

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

   5. Taylor expanded in im around 0 90.6%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \cdot \cos re\right) \]
   6. Step-by-step derivation

      1. unpow2 90.6%

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

   7. Applied egg-rr 90.6%

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

   if 1e12 < im < 1.06e44

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

   6. Taylor expanded in re around 0 61.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right)} \]

   7. Taylor expanded in re around inf 61.3%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r* 61.3%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{4}}\right) \]

      2. *-commutative 61.3%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot {re}^{4}\right) \]

      3. associate-*l* 61.3%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)}\right) \]

   9. Simplified 61.3%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)}\right) \]

   if 1.06e44 < im < 8.1999999999999999e102

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 81.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1000000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 480000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{+256} \lor \neg \left(im \leq 6.5 \cdot 10^{+294}\right):\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 480000000.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (or (<= im 4.7e+256) (not (<= im 6.5e+294)))
     (* 0.5 (* -0.0003968253968253968 (pow im 7.0)))
     (* 0.5 (* im (+ -2.0 (pow re 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 480000000.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if ((im <= 4.7e+256) || !(im <= 6.5e+294)) {
		tmp = 0.5 * (-0.0003968253968253968 * pow(im, 7.0));
	} else {
		tmp = 0.5 * (im * (-2.0 + pow(re, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 480000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else if ((im <= 4.7d+256) .or. (.not. (im <= 6.5d+294))) then
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (im ** 7.0d0))
    else
        tmp = 0.5d0 * (im * ((-2.0d0) + (re ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 480000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if ((im <= 4.7e+256) || !(im <= 6.5e+294)) {
		tmp = 0.5 * (-0.0003968253968253968 * Math.pow(im, 7.0));
	} else {
		tmp = 0.5 * (im * (-2.0 + Math.pow(re, 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 480000000.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif (im <= 4.7e+256) or not (im <= 6.5e+294):
		tmp = 0.5 * (-0.0003968253968253968 * math.pow(im, 7.0))
	else:
		tmp = 0.5 * (im * (-2.0 + math.pow(re, 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 480000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif ((im <= 4.7e+256) || !(im <= 6.5e+294))
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * (im ^ 7.0)));
	else
		tmp = Float64(0.5 * Float64(im * Float64(-2.0 + (re ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 480000000.0)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	elseif ((im <= 4.7e+256) || ~((im <= 6.5e+294)))
		tmp = 0.5 * (-0.0003968253968253968 * (im ^ 7.0));
	else
		tmp = 0.5 * (im * (-2.0 + (re ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 480000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 4.7e+256], N[Not[LessEqual[im, 6.5e+294]], $MachinePrecision]], N[(0.5 * N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.8e8

   1. Initial program 35.5%

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

      1. /-rgt-identity 35.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 35.5%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 35.5%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 35.5%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 35.5%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 35.5%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 35.5%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 35.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 35.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 35.5%

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

      11. cos-neg 35.5%

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

   3. Simplified 35.5%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 4.8e8 < im < 4.69999999999999967e256 or 6.49999999999999965e294 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 90.8%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 90.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 69.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]

   if 4.69999999999999967e256 < im < 6.49999999999999965e294

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 8.4%

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

   6. Taylor expanded in re around 0 61.7%

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

      1. +-commutative 61.7%

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

      2. *-commutative 61.7%

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

      3. distribute-lft-out 61.7%

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

   8. Simplified 61.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 480000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 4.7 \cdot 10^{+256} \lor \neg \left(im \leq 6.5 \cdot 10^{+294}\right):\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 980000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+260}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+294}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 980000000000.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 1.2e+260)
     (* 0.5 (* im (- (* -0.0003968253968253968 (pow im 6.0)) 2.0)))
     (if (<= im 6.5e+294)
       (* 0.5 (* im (+ -2.0 (pow re 2.0))))
       (* 0.5 (* -0.0003968253968253968 (pow im 7.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 980000000000.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 1.2e+260) {
		tmp = 0.5 * (im * ((-0.0003968253968253968 * pow(im, 6.0)) - 2.0));
	} else if (im <= 6.5e+294) {
		tmp = 0.5 * (im * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * pow(im, 7.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 980000000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else if (im <= 1.2d+260) then
        tmp = 0.5d0 * (im * (((-0.0003968253968253968d0) * (im ** 6.0d0)) - 2.0d0))
    else if (im <= 6.5d+294) then
        tmp = 0.5d0 * (im * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (im ** 7.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 980000000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 1.2e+260) {
		tmp = 0.5 * (im * ((-0.0003968253968253968 * Math.pow(im, 6.0)) - 2.0));
	} else if (im <= 6.5e+294) {
		tmp = 0.5 * (im * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * Math.pow(im, 7.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 980000000000.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 1.2e+260:
		tmp = 0.5 * (im * ((-0.0003968253968253968 * math.pow(im, 6.0)) - 2.0))
	elif im <= 6.5e+294:
		tmp = 0.5 * (im * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = 0.5 * (-0.0003968253968253968 * math.pow(im, 7.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 980000000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 1.2e+260)
		tmp = Float64(0.5 * Float64(im * Float64(Float64(-0.0003968253968253968 * (im ^ 6.0)) - 2.0)));
	elseif (im <= 6.5e+294)
		tmp = Float64(0.5 * Float64(im * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * (im ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 980000000000.0)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	elseif (im <= 1.2e+260)
		tmp = 0.5 * (im * ((-0.0003968253968253968 * (im ^ 6.0)) - 2.0));
	elseif (im <= 6.5e+294)
		tmp = 0.5 * (im * (-2.0 + (re ^ 2.0)));
	else
		tmp = 0.5 * (-0.0003968253968253968 * (im ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 980000000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.2e+260], N[(0.5 * N[(im * N[(N[(-0.0003968253968253968 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.5e+294], N[(0.5 * N[(im * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 9.8e11

   1. Initial program 35.5%

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

      1. /-rgt-identity 35.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 35.5%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 35.5%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 35.5%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 35.5%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 35.5%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 35.5%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 35.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 35.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 35.5%

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

      11. cos-neg 35.5%

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

   3. Simplified 35.5%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 9.8e11 < im < 1.20000000000000005e260

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 90.8%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 90.8%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(\color{blue}{-0.0003968253968253968 \cdot {im}^{6}} - 2\right)\right) \cdot \cos re\right) \]

   7. Taylor expanded in re around 0 69.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)} \]

   if 1.20000000000000005e260 < im < 6.49999999999999965e294

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 8.6%

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

   6. Taylor expanded in re around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. *-commutative 68.5%

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

      3. distribute-lft-out 68.5%

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

   8. Simplified 68.5%

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

   if 6.49999999999999965e294 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 980000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+260}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+294}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 980000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 980000000000.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (* 0.5 (* -0.0003968253968253968 (pow im 7.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 980000000000.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * pow(im, 7.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 980000000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (im ** 7.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 980000000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * Math.pow(im, 7.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 980000000000.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	else:
		tmp = 0.5 * (-0.0003968253968253968 * math.pow(im, 7.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 980000000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	else
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * (im ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 980000000000.0)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	else
		tmp = 0.5 * (-0.0003968253968253968 * (im ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 980000000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.8e11

   1. Initial program 35.5%

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

      1. /-rgt-identity 35.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 35.5%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 35.5%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 35.5%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 35.5%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 35.5%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 35.5%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 35.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 35.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 35.5%

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

      11. cos-neg 35.5%

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

   3. Simplified 35.5%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 9.8e11 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 92.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 92.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 64.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 980000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.0)
   (* 0.5 (* im -2.0))
   (* 0.5 (* -0.0003968253968253968 (pow im 7.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.0) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * pow(im, 7.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.0d0) then
        tmp = 0.5d0 * (im * (-2.0d0))
    else
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (im ** 7.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.0) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * Math.pow(im, 7.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.0:
		tmp = 0.5 * (im * -2.0)
	else:
		tmp = 0.5 * (-0.0003968253968253968 * math.pow(im, 7.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.0)
		tmp = Float64(0.5 * Float64(im * -2.0));
	else
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * (im ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.0)
		tmp = 0.5 * (im * -2.0);
	else
		tmp = 0.5 * (-0.0003968253968253968 * (im ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.0], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4

   1. Initial program 34.5%

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

      1. /-rgt-identity 34.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 34.5%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 34.5%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 34.5%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 34.5%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 34.5%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 34.5%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 34.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 34.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 34.5%

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

      11. cos-neg 34.5%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in im around 0 71.8%

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

   6. Taylor expanded in re around 0 37.2%

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

      1. *-commutative 37.2%

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

   8. Simplified 37.2%

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

   if 4 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 88.2%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 88.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 61.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 30.0% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot -2\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* im -2.0)))

C

double code(double re, double im) {
	return 0.5 * (im * -2.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (im * (-2.0d0))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (im * -2.0);
}

Python

def code(re, im):
	return 0.5 * (im * -2.0)

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * -2.0))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im * -2.0);
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

   1. /-rgt-identity 50.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 50.9%

      \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. associate-*l/ 50.9%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   4. cos-neg 50.9%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

   5. associate-*l* 50.9%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

   6. associate-*r/ 50.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   7. exp-0 50.9%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 50.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   9. *-commutative 50.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

   10. neg-sub0 50.9%

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

   11. cos-neg 50.9%

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

3. Simplified 50.9%

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

5. Taylor expanded in im around 0 55.2%

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

6. Taylor expanded in re around 0 28.9%

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

   1. *-commutative 28.9%

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

8. Simplified 28.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot -2\right)} \]
9. Add Preprocessing

Developer target: 99.8% accurate, 0.7× 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 2024106 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (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))))