math.sin on complex, imaginary part

Percentage Accurate: 53.9% → 99.1%

Time: 38.9s

Alternatives: 10

Speedup: 2.5×

• 49.2% 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: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

   1. /-rgt-identity 53.5%

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

   2. exp-0 53.5%

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

   3. associate-*l/ 53.5%

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

   4. cos-neg 53.5%

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

   5. associate-*l* 53.5%

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

   6. associate-*r/ 53.5%

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

   7. exp-0 53.5%

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

   8. /-rgt-identity 53.5%

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

   9. *-commutative 53.5%

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

   10. neg-sub0 53.5%

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

   11. cos-neg 53.5%

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

3. Simplified 53.5%

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

5. Taylor expanded in im around 0 52.4%

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

   1. log1p-expm1-u 99.3%

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

   2. associate-*l* 99.3%

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

7. Applied egg-rr 99.3%

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

Alternative 2: 68.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+228}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+238}:\\ \;\;\;\;0.5 \cdot \sqrt{{im}^{10} \cdot 0.0002777777777777778}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 460.0)
   (* 0.5 (* (cos re) (* -2.0 im)))
   (if (<= im 8.5e+228)
     (* 0.5 (log1p (expm1 (* -2.0 im))))
     (if (<= im 1.25e+238)
       (* 0.5 (sqrt (* (pow im 10.0) 0.0002777777777777778)))
       (* 0.5 (* -0.016666666666666666 (pow im 5.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 460.0) {
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	} else if (im <= 8.5e+228) {
		tmp = 0.5 * log1p(expm1((-2.0 * im)));
	} else if (im <= 1.25e+238) {
		tmp = 0.5 * sqrt((pow(im, 10.0) * 0.0002777777777777778));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * pow(im, 5.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 460.0) {
		tmp = 0.5 * (Math.cos(re) * (-2.0 * im));
	} else if (im <= 8.5e+228) {
		tmp = 0.5 * Math.log1p(Math.expm1((-2.0 * im)));
	} else if (im <= 1.25e+238) {
		tmp = 0.5 * Math.sqrt((Math.pow(im, 10.0) * 0.0002777777777777778));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * Math.pow(im, 5.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 460.0:
		tmp = 0.5 * (math.cos(re) * (-2.0 * im))
	elif im <= 8.5e+228:
		tmp = 0.5 * math.log1p(math.expm1((-2.0 * im)))
	elif im <= 1.25e+238:
		tmp = 0.5 * math.sqrt((math.pow(im, 10.0) * 0.0002777777777777778))
	else:
		tmp = 0.5 * (-0.016666666666666666 * math.pow(im, 5.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 460.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im)));
	elseif (im <= 8.5e+228)
		tmp = Float64(0.5 * log1p(expm1(Float64(-2.0 * im))));
	elseif (im <= 1.25e+238)
		tmp = Float64(0.5 * sqrt(Float64((im ^ 10.0) * 0.0002777777777777778)));
	else
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * (im ^ 5.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 460.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.5e+228], N[(0.5 * N[Log[1 + N[(Exp[N[(-2.0 * im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.25e+238], N[(0.5 * N[Sqrt[N[(N[Power[im, 10.0], $MachinePrecision] * 0.0002777777777777778), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 460

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 68.1%

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

   if 460 < im < 8.5000000000000002e228

   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 4.5%

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

      1. log1p-expm1-u 100.0%

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

      2. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 75.0%

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

   if 8.5000000000000002e228 < im < 1.24999999999999999e238

   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%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in re around 0 0.0%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot {im}^{5}\right)} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{-0.016666666666666666 \cdot {im}^{5}} \cdot \sqrt{-0.016666666666666666 \cdot {im}^{5}}\right)} \]

       2. sqrt-unprod 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\sqrt{\left(-0.016666666666666666 \cdot {im}^{5}\right) \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)}} \]

       3. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left({im}^{5} \cdot -0.016666666666666666\right)} \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)} \]

       4. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \sqrt{\left({im}^{5} \cdot -0.016666666666666666\right) \cdot \color{blue}{\left({im}^{5} \cdot -0.016666666666666666\right)}} \]

       5. swap-sqr 100.0%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left({im}^{5} \cdot {im}^{5}\right) \cdot \left(-0.016666666666666666 \cdot -0.016666666666666666\right)}} \]

       6. pow-prod-up 100.0%

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

       7. metadata-eval 100.0%

          \[\leadsto 0.5 \cdot \sqrt{{im}^{\color{blue}{10}} \cdot \left(-0.016666666666666666 \cdot -0.016666666666666666\right)} \]

       8. metadata-eval 100.0%

          \[\leadsto 0.5 \cdot \sqrt{{im}^{10} \cdot \color{blue}{0.0002777777777777778}} \]

   13. Applied egg-rr 100.0%

       \[\leadsto 0.5 \cdot \color{blue}{\sqrt{{im}^{10} \cdot 0.0002777777777777778}} \]

   if 1.24999999999999999e238 < 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%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in re around 0 90.0%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot {im}^{5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+228}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+238}:\\ \;\;\;\;0.5 \cdot \sqrt{{im}^{10} \cdot 0.0002777777777777778}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 490:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333\right) + -2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 490.0)
   (*
    0.5
    (* (cos re) (+ (* im (* (pow im 2.0) -0.3333333333333333)) (* -2.0 im))))
   (if (<= im 4.5e+61)
     (* 0.5 (log1p (expm1 (* -2.0 im))))
     (* 0.5 (* (cos re) (* -0.016666666666666666 (pow im 5.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 490.0) {
		tmp = 0.5 * (cos(re) * ((im * (pow(im, 2.0) * -0.3333333333333333)) + (-2.0 * im)));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * log1p(expm1((-2.0 * im)));
	} else {
		tmp = 0.5 * (cos(re) * (-0.016666666666666666 * pow(im, 5.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 490.0) {
		tmp = 0.5 * (Math.cos(re) * ((im * (Math.pow(im, 2.0) * -0.3333333333333333)) + (-2.0 * im)));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * Math.log1p(Math.expm1((-2.0 * im)));
	} else {
		tmp = 0.5 * (Math.cos(re) * (-0.016666666666666666 * Math.pow(im, 5.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 490.0:
		tmp = 0.5 * (math.cos(re) * ((im * (math.pow(im, 2.0) * -0.3333333333333333)) + (-2.0 * im)))
	elif im <= 4.5e+61:
		tmp = 0.5 * math.log1p(math.expm1((-2.0 * im)))
	else:
		tmp = 0.5 * (math.cos(re) * (-0.016666666666666666 * math.pow(im, 5.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 490.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(im * Float64((im ^ 2.0) * -0.3333333333333333)) + Float64(-2.0 * im))));
	elseif (im <= 4.5e+61)
		tmp = Float64(0.5 * log1p(expm1(Float64(-2.0 * im))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-0.016666666666666666 * (im ^ 5.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 490.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(im * N[(N[Power[im, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.5e+61], N[(0.5 * N[Log[1 + N[(Exp[N[(-2.0 * im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 490

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 89.6%

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

      1. sub-neg 89.6%

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

      2. metadata-eval 89.6%

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

      3. distribute-rgt-in 89.6%

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

      4. *-commutative 89.6%

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

   7. Applied egg-rr 89.6%

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

   if 490 < im < 4.5e61

   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 3.4%

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

      1. log1p-expm1-u 100.0%

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

      2. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 87.5%

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

   if 4.5e61 < 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%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 490:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333\right) + -2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 500:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 500.0)
   (* 0.5 (* (cos re) (* im (- (* (pow im 2.0) -0.3333333333333333) 2.0))))
   (if (<= im 4.5e+61)
     (* 0.5 (log1p (expm1 (* -2.0 im))))
     (* 0.5 (* (cos re) (* -0.016666666666666666 (pow im 5.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 500.0) {
		tmp = 0.5 * (cos(re) * (im * ((pow(im, 2.0) * -0.3333333333333333) - 2.0)));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * log1p(expm1((-2.0 * im)));
	} else {
		tmp = 0.5 * (cos(re) * (-0.016666666666666666 * pow(im, 5.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 500.0) {
		tmp = 0.5 * (Math.cos(re) * (im * ((Math.pow(im, 2.0) * -0.3333333333333333) - 2.0)));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * Math.log1p(Math.expm1((-2.0 * im)));
	} else {
		tmp = 0.5 * (Math.cos(re) * (-0.016666666666666666 * Math.pow(im, 5.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 500.0:
		tmp = 0.5 * (math.cos(re) * (im * ((math.pow(im, 2.0) * -0.3333333333333333) - 2.0)))
	elif im <= 4.5e+61:
		tmp = 0.5 * math.log1p(math.expm1((-2.0 * im)))
	else:
		tmp = 0.5 * (math.cos(re) * (-0.016666666666666666 * math.pow(im, 5.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 500.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * Float64(Float64((im ^ 2.0) * -0.3333333333333333) - 2.0))));
	elseif (im <= 4.5e+61)
		tmp = Float64(0.5 * log1p(expm1(Float64(-2.0 * im))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-0.016666666666666666 * (im ^ 5.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 500.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * N[(N[(N[Power[im, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.5e+61], N[(0.5 * N[Log[1 + N[(Exp[N[(-2.0 * im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 500

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 89.6%

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

   if 500 < im < 4.5e61

   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 3.4%

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

      1. log1p-expm1-u 100.0%

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

      2. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 87.5%

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

   if 4.5e61 < 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%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 500:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 460.0)
   (* 0.5 (* (cos re) (* -2.0 im)))
   (if (<= im 4.5e+61)
     (* 0.5 (log1p (expm1 (* -2.0 im))))
     (* 0.5 (* (cos re) (* -0.016666666666666666 (pow im 5.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 460.0) {
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * log1p(expm1((-2.0 * im)));
	} else {
		tmp = 0.5 * (cos(re) * (-0.016666666666666666 * pow(im, 5.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 460.0) {
		tmp = 0.5 * (Math.cos(re) * (-2.0 * im));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * Math.log1p(Math.expm1((-2.0 * im)));
	} else {
		tmp = 0.5 * (Math.cos(re) * (-0.016666666666666666 * Math.pow(im, 5.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 460.0:
		tmp = 0.5 * (math.cos(re) * (-2.0 * im))
	elif im <= 4.5e+61:
		tmp = 0.5 * math.log1p(math.expm1((-2.0 * im)))
	else:
		tmp = 0.5 * (math.cos(re) * (-0.016666666666666666 * math.pow(im, 5.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 460.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im)));
	elseif (im <= 4.5e+61)
		tmp = Float64(0.5 * log1p(expm1(Float64(-2.0 * im))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-0.016666666666666666 * (im ^ 5.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 460.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.5e+61], N[(0.5 * N[Log[1 + N[(Exp[N[(-2.0 * im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 460

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 68.1%

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

   if 460 < im < 4.5e61

   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 3.4%

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

      1. log1p-expm1-u 100.0%

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

      2. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 87.5%

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

   if 4.5e61 < 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%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 600:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+228}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+238}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 600.0)
   (* 0.5 (* (cos re) (* -2.0 im)))
   (if (<= im 8.5e+228)
     (* 0.5 (log1p (expm1 (* -2.0 im))))
     (if (<= im 1.25e+238)
       (* 0.5 (* im (fma re re -2.0)))
       (* 0.5 (* -0.016666666666666666 (pow im 5.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 600.0) {
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	} else if (im <= 8.5e+228) {
		tmp = 0.5 * log1p(expm1((-2.0 * im)));
	} else if (im <= 1.25e+238) {
		tmp = 0.5 * (im * fma(re, re, -2.0));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * pow(im, 5.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 600.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im)));
	elseif (im <= 8.5e+228)
		tmp = Float64(0.5 * log1p(expm1(Float64(-2.0 * im))));
	elseif (im <= 1.25e+238)
		tmp = Float64(0.5 * Float64(im * fma(re, re, -2.0)));
	else
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * (im ^ 5.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 600.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.5e+228], N[(0.5 * N[Log[1 + N[(Exp[N[(-2.0 * im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.25e+238], N[(0.5 * N[(im * N[(re * re + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 600

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 68.1%

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

   if 600 < im < 8.5000000000000002e228

   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 4.5%

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

      1. log1p-expm1-u 100.0%

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

      2. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 75.0%

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

   if 8.5000000000000002e228 < im < 1.24999999999999999e238

   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 6.3%

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

      1. log1p-expm1-u 100.0%

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

      2. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. unpow2 100.0%

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

      5. fma-undefine 100.0%

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

   10. Simplified 100.0%

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

   if 1.24999999999999999e238 < 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%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in re around 0 90.0%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot {im}^{5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 600:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+228}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+238}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4200000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+228} \lor \neg \left(im \leq 1.25 \cdot 10^{+238}\right):\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4200000000000.0)
   (* 0.5 (* (cos re) (* -2.0 im)))
   (if (or (<= im 8e+228) (not (<= im 1.25e+238)))
     (* 0.5 (* -0.016666666666666666 (pow im 5.0)))
     (* 0.5 (* im (fma re re -2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4200000000000.0) {
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	} else if ((im <= 8e+228) || !(im <= 1.25e+238)) {
		tmp = 0.5 * (-0.016666666666666666 * pow(im, 5.0));
	} else {
		tmp = 0.5 * (im * fma(re, re, -2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4200000000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im)));
	elseif ((im <= 8e+228) || !(im <= 1.25e+238))
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * (im ^ 5.0)));
	else
		tmp = Float64(0.5 * Float64(im * fma(re, re, -2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4200000000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 8e+228], N[Not[LessEqual[im, 1.25e+238]], $MachinePrecision]], N[(0.5 * N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(re * re + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.2e12

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 68.1%

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

   if 4.2e12 < im < 7.9999999999999994e228 or 1.24999999999999999e238 < 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 87.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-*r* 87.8%

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

      4. *-commutative 87.8%

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

      5. fma-undefine 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in im around inf 87.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 87.8%

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

      2. *-commutative 87.8%

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

   10. Simplified 87.8%

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

   11. Taylor expanded in re around 0 66.8%

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

   if 7.9999999999999994e228 < im < 1.24999999999999999e238

   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 6.3%

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

      1. log1p-expm1-u 100.0%

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

      2. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. unpow2 100.0%

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

      5. fma-undefine 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4200000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+228} \lor \neg \left(im \leq 1.25 \cdot 10^{+238}\right):\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.3:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+228} \lor \neg \left(im \leq 1.85 \cdot 10^{+238}\right):\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.3)
   (* 0.5 (* -2.0 im))
   (if (or (<= im 8.5e+228) (not (<= im 1.85e+238)))
     (* 0.5 (* -0.016666666666666666 (pow im 5.0)))
     (* 0.5 (* im (fma re re -2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.3) {
		tmp = 0.5 * (-2.0 * im);
	} else if ((im <= 8.5e+228) || !(im <= 1.85e+238)) {
		tmp = 0.5 * (-0.016666666666666666 * pow(im, 5.0));
	} else {
		tmp = 0.5 * (im * fma(re, re, -2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.3)
		tmp = Float64(0.5 * Float64(-2.0 * im));
	elseif ((im <= 8.5e+228) || !(im <= 1.85e+238))
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * (im ^ 5.0)));
	else
		tmp = Float64(0.5 * Float64(im * fma(re, re, -2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.3], N[(0.5 * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 8.5e+228], N[Not[LessEqual[im, 1.85e+238]], $MachinePrecision]], N[(0.5 * N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(re * re + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.2999999999999998

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 68.1%

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

   6. Taylor expanded in re around 0 36.5%

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

   if 3.2999999999999998 < im < 8.5000000000000002e228 or 1.85e238 < 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 87.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-*r* 87.8%

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

      4. *-commutative 87.8%

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

      5. fma-undefine 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in im around inf 87.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 87.8%

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

      2. *-commutative 87.8%

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

   10. Simplified 87.8%

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

   11. Taylor expanded in re around 0 66.8%

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

   if 8.5000000000000002e228 < im < 1.85e238

   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 6.3%

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

      1. log1p-expm1-u 100.0%

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

      2. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. unpow2 100.0%

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

      5. fma-undefine 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.3:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+228} \lor \neg \left(im \leq 1.85 \cdot 10^{+238}\right):\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.3:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.3)
   (* 0.5 (* -2.0 im))
   (* 0.5 (* -0.016666666666666666 (pow im 5.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.3) {
		tmp = 0.5 * (-2.0 * im);
	} else {
		tmp = 0.5 * (-0.016666666666666666 * pow(im, 5.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.3d0) then
        tmp = 0.5d0 * ((-2.0d0) * im)
    else
        tmp = 0.5d0 * ((-0.016666666666666666d0) * (im ** 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.3) {
		tmp = 0.5 * (-2.0 * im);
	} else {
		tmp = 0.5 * (-0.016666666666666666 * Math.pow(im, 5.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.3:
		tmp = 0.5 * (-2.0 * im)
	else:
		tmp = 0.5 * (-0.016666666666666666 * math.pow(im, 5.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.3)
		tmp = Float64(0.5 * Float64(-2.0 * im));
	else
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * (im ^ 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.3)
		tmp = 0.5 * (-2.0 * im);
	else
		tmp = 0.5 * (-0.016666666666666666 * (im ^ 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.3], N[(0.5 * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.2999999999999998

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 68.1%

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

   6. Taylor expanded in re around 0 36.5%

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

   if 3.2999999999999998 < 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%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 88.2%

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

      2. +-commutative 88.2%

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

      3. associate-*r* 88.2%

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

      4. *-commutative 88.2%

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

      5. fma-undefine 88.2%

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

   7. Simplified 88.2%

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

   8. Taylor expanded in im around inf 88.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 88.2%

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

      2. *-commutative 88.2%

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

   10. Simplified 88.2%

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

   11. Taylor expanded in re around 0 64.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot {im}^{5}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 30.2% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

   1. /-rgt-identity 53.5%

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

   2. exp-0 53.5%

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

   3. associate-*l/ 53.5%

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

   4. cos-neg 53.5%

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

   5. associate-*l* 53.5%

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

   6. associate-*r/ 53.5%

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

   7. exp-0 53.5%

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

   8. /-rgt-identity 53.5%

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

   9. *-commutative 53.5%

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

   10. neg-sub0 53.5%

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

   11. cos-neg 53.5%

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

3. Simplified 53.5%

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

5. Taylor expanded in im around 0 52.4%

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

6. Taylor expanded in re around 0 28.3%

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

Developer Target 1: 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 2024115 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (cos re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (cos re)) (- (exp (- 0 im)) (exp im)))))

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