math.sin on complex, imaginary part

Percentage Accurate: 55.0% → 98.9%

Time: 9.6s

Alternatives: 10

Speedup: 2.8×

• 50.1% 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: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

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 52.0%

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

   1. cos-neg 52.0%

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

   2. sub-neg 52.0%

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

   3. neg-sub0 52.0%

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

   4. remove-double-neg 52.0%

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

   5. remove-double-neg 52.0%

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

   6. sub0-neg 52.0%

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

   7. distribute-neg-in 52.0%

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

   8. +-commutative 52.0%

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

   9. sub-neg 52.0%

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

   10. associate-*l* 52.0%

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

   11. sub-neg 52.0%

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

   12. +-commutative 52.0%

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

   13. distribute-neg-in 52.0%

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

3. Simplified 52.0%

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

5. Taylor expanded in im around 0 53.7%

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

   1. log1p-expm1-u 99.2%

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

   2. associate-*l* 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 2: 89.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 435:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 435.0)
   (* 0.5 (* (cos re) (+ (* -2.0 im) (* -0.3333333333333333 (pow im 3.0)))))
   (if (<= im 5.5e+102)
     (* 0.5 (log1p (expm1 (* -2.0 im))))
     (* (pow im 3.0) (* (cos re) -0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 435.0) {
		tmp = 0.5 * (cos(re) * ((-2.0 * im) + (-0.3333333333333333 * pow(im, 3.0))));
	} else if (im <= 5.5e+102) {
		tmp = 0.5 * log1p(expm1((-2.0 * im)));
	} else {
		tmp = pow(im, 3.0) * (cos(re) * -0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 435.0) {
		tmp = 0.5 * (Math.cos(re) * ((-2.0 * im) + (-0.3333333333333333 * Math.pow(im, 3.0))));
	} else if (im <= 5.5e+102) {
		tmp = 0.5 * Math.log1p(Math.expm1((-2.0 * im)));
	} else {
		tmp = Math.pow(im, 3.0) * (Math.cos(re) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 435.0:
		tmp = 0.5 * (math.cos(re) * ((-2.0 * im) + (-0.3333333333333333 * math.pow(im, 3.0))))
	elif im <= 5.5e+102:
		tmp = 0.5 * math.log1p(math.expm1((-2.0 * im)))
	else:
		tmp = math.pow(im, 3.0) * (math.cos(re) * -0.16666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 435.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(-2.0 * im) + Float64(-0.3333333333333333 * (im ^ 3.0)))));
	elseif (im <= 5.5e+102)
		tmp = Float64(0.5 * log1p(expm1(Float64(-2.0 * im))));
	else
		tmp = Float64((im ^ 3.0) * Float64(cos(re) * -0.16666666666666666));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 435.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(-2.0 * im), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.5e+102], N[(0.5 * N[Log[1 + N[(Exp[N[(-2.0 * im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 3.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 435

   1. Initial program 34.2%

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

      1. cos-neg 34.2%

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

      2. sub-neg 34.2%

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

      3. neg-sub0 34.2%

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

      4. remove-double-neg 34.2%

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

      5. remove-double-neg 34.2%

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

      6. sub0-neg 34.2%

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

      7. distribute-neg-in 34.2%

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

      8. +-commutative 34.2%

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

      9. sub-neg 34.2%

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

      10. associate-*l* 34.2%

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

      11. sub-neg 34.2%

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

      12. +-commutative 34.2%

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

      13. distribute-neg-in 34.2%

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

   3. Simplified 34.2%

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

   5. Taylor expanded in im around 0 90.9%

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

   if 435 < im < 5.49999999999999981e102

   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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   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 3.5%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im\right)} \cdot \cos re\right) \]
   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 69.2%

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

   if 5.49999999999999981e102 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left({im}^{3} \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in im around 0 100.0%

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

       1. *-commutative 100.0%

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

       2. *-commutative 100.0%

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

       3. *-commutative 100.0%

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

       4. associate-*l* 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 435:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.8e-21)
   (* 0.5 (* (cos re) (* -2.0 im)))
   (if (<= im 5.5e+102)
     (* 0.5 (log1p (expm1 (* -2.0 im))))
     (* (pow im 3.0) (* (cos re) -0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.8e-21) {
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	} else if (im <= 5.5e+102) {
		tmp = 0.5 * log1p(expm1((-2.0 * im)));
	} else {
		tmp = pow(im, 3.0) * (cos(re) * -0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.8e-21) {
		tmp = 0.5 * (Math.cos(re) * (-2.0 * im));
	} else if (im <= 5.5e+102) {
		tmp = 0.5 * Math.log1p(Math.expm1((-2.0 * im)));
	} else {
		tmp = Math.pow(im, 3.0) * (Math.cos(re) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.8e-21:
		tmp = 0.5 * (math.cos(re) * (-2.0 * im))
	elif im <= 5.5e+102:
		tmp = 0.5 * math.log1p(math.expm1((-2.0 * im)))
	else:
		tmp = math.pow(im, 3.0) * (math.cos(re) * -0.16666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.8e-21)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im)));
	elseif (im <= 5.5e+102)
		tmp = Float64(0.5 * log1p(expm1(Float64(-2.0 * im))));
	else
		tmp = Float64((im ^ 3.0) * Float64(cos(re) * -0.16666666666666666));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.8e-21], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.5e+102], N[(0.5 * N[Log[1 + N[(Exp[N[(-2.0 * im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 3.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.80000000000000004e-21

   1. Initial program 33.6%

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

      1. cos-neg 33.6%

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

      2. sub-neg 33.6%

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

      3. neg-sub0 33.6%

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

      4. remove-double-neg 33.6%

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

      5. remove-double-neg 33.6%

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

      6. sub0-neg 33.6%

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

      7. distribute-neg-in 33.6%

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

      8. +-commutative 33.6%

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

      9. sub-neg 33.6%

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

      10. associate-*l* 33.6%

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

      11. sub-neg 33.6%

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

      12. +-commutative 33.6%

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

      13. distribute-neg-in 33.6%

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

   3. Simplified 33.6%

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

   5. Taylor expanded in im around 0 71.9%

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

   if 2.80000000000000004e-21 < im < 5.49999999999999981e102

   1. Initial program 95.0%

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

      1. cos-neg 95.0%

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

      2. sub-neg 95.0%

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

      3. neg-sub0 95.0%

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

      4. remove-double-neg 95.0%

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

      5. remove-double-neg 95.0%

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

      6. sub0-neg 95.0%

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

      7. distribute-neg-in 95.0%

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

      8. +-commutative 95.0%

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

      9. sub-neg 95.0%

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

      10. associate-*l* 95.0%

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

      11. sub-neg 95.0%

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

      12. +-commutative 95.0%

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

      13. distribute-neg-in 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in im around 0 10.6%

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

      1. log1p-expm1-u 94.2%

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

      2. associate-*l* 94.2%

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

   7. Applied egg-rr 94.2%

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

   8. Taylor expanded in re around 0 67.6%

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

   if 5.49999999999999981e102 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left({im}^{3} \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in im around 0 100.0%

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

       1. *-commutative 100.0%

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

       2. *-commutative 100.0%

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

       3. *-commutative 100.0%

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

       4. associate-*l* 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.32 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{{im}^{6} \cdot 0.027777777777777776}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.32e+50)
   (* 0.5 (* (cos re) (* -2.0 im)))
   (if (<= im 1.5e+110)
     (sqrt (* (pow im 6.0) 0.027777777777777776))
     (* 0.5 (+ (* -2.0 im) (* -0.3333333333333333 (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.32e+50) {
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	} else if (im <= 1.5e+110) {
		tmp = sqrt((pow(im, 6.0) * 0.027777777777777776));
	} else {
		tmp = 0.5 * ((-2.0 * im) + (-0.3333333333333333 * pow(im, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.32d+50) then
        tmp = 0.5d0 * (cos(re) * ((-2.0d0) * im))
    else if (im <= 1.5d+110) then
        tmp = sqrt(((im ** 6.0d0) * 0.027777777777777776d0))
    else
        tmp = 0.5d0 * (((-2.0d0) * im) + ((-0.3333333333333333d0) * (im ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.32e+50) {
		tmp = 0.5 * (Math.cos(re) * (-2.0 * im));
	} else if (im <= 1.5e+110) {
		tmp = Math.sqrt((Math.pow(im, 6.0) * 0.027777777777777776));
	} else {
		tmp = 0.5 * ((-2.0 * im) + (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.32e+50:
		tmp = 0.5 * (math.cos(re) * (-2.0 * im))
	elif im <= 1.5e+110:
		tmp = math.sqrt((math.pow(im, 6.0) * 0.027777777777777776))
	else:
		tmp = 0.5 * ((-2.0 * im) + (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.32e+50)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im)));
	elseif (im <= 1.5e+110)
		tmp = sqrt(Float64((im ^ 6.0) * 0.027777777777777776));
	else
		tmp = Float64(0.5 * Float64(Float64(-2.0 * im) + Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.32e+50)
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	elseif (im <= 1.5e+110)
		tmp = sqrt(((im ^ 6.0) * 0.027777777777777776));
	else
		tmp = 0.5 * ((-2.0 * im) + (-0.3333333333333333 * (im ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.32e+50], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.5e+110], N[Sqrt[N[(N[Power[im, 6.0], $MachinePrecision] * 0.027777777777777776), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(N[(-2.0 * im), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.3199999999999999e50

   1. Initial program 38.5%

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

      1. cos-neg 38.5%

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

      2. sub-neg 38.5%

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

      3. neg-sub0 38.5%

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

      4. remove-double-neg 38.5%

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

      5. remove-double-neg 38.5%

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

      6. sub0-neg 38.5%

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

      7. distribute-neg-in 38.5%

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

      8. +-commutative 38.5%

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

      9. sub-neg 38.5%

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

      10. associate-*l* 38.5%

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

      11. sub-neg 38.5%

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

      12. +-commutative 38.5%

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

      13. distribute-neg-in 38.5%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in im around 0 67.1%

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

   if 1.3199999999999999e50 < im < 1.50000000000000004e110

   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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   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 19.4%

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

   6. Taylor expanded in im around inf 19.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left({im}^{3} \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 19.4%

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

      2. associate-*l* 19.4%

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

   8. Simplified 19.4%

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

   9. Taylor expanded in re around 0 10.6%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative 10.6%

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

   11. Simplified 10.6%

       \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 33.6%

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

       3. swap-sqr 33.6%

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

       4. pow-prod-up 33.6%

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

       5. metadata-eval 33.6%

          \[\leadsto \sqrt{{im}^{\color{blue}{6}} \cdot \left(-0.16666666666666666 \cdot -0.16666666666666666\right)} \]

       6. metadata-eval 33.6%

          \[\leadsto \sqrt{{im}^{6} \cdot \color{blue}{0.027777777777777776}} \]

   13. Applied egg-rr 33.6%

       \[\leadsto \color{blue}{\sqrt{{im}^{6} \cdot 0.027777777777777776}} \]

   if 1.50000000000000004e110 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 68.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.32 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{{im}^{6} \cdot 0.027777777777777776}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.8e-21)
   (* 0.5 (* (cos re) (* -2.0 im)))
   (* 0.5 (log1p (expm1 (* -2.0 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.8e-21) {
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	} else {
		tmp = 0.5 * log1p(expm1((-2.0 * im)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.8e-21) {
		tmp = 0.5 * (Math.cos(re) * (-2.0 * im));
	} else {
		tmp = 0.5 * Math.log1p(Math.expm1((-2.0 * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.8e-21:
		tmp = 0.5 * (math.cos(re) * (-2.0 * im))
	else:
		tmp = 0.5 * math.log1p(math.expm1((-2.0 * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.8e-21)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im)));
	else
		tmp = Float64(0.5 * log1p(expm1(Float64(-2.0 * im))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.8e-21], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Log[1 + N[(Exp[N[(-2.0 * im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.80000000000000004e-21

   1. Initial program 33.6%

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

      1. cos-neg 33.6%

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

      2. sub-neg 33.6%

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

      3. neg-sub0 33.6%

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

      4. remove-double-neg 33.6%

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

      5. remove-double-neg 33.6%

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

      6. sub0-neg 33.6%

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

      7. distribute-neg-in 33.6%

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

      8. +-commutative 33.6%

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

      9. sub-neg 33.6%

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

      10. associate-*l* 33.6%

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

      11. sub-neg 33.6%

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

      12. +-commutative 33.6%

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

      13. distribute-neg-in 33.6%

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

   3. Simplified 33.6%

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

   5. Taylor expanded in im around 0 71.9%

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

   if 2.80000000000000004e-21 < im

   1. Initial program 98.0%

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

      1. cos-neg 98.0%

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

      2. sub-neg 98.0%

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

      3. neg-sub0 98.0%

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

      4. remove-double-neg 98.0%

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

      5. remove-double-neg 98.0%

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

      6. sub0-neg 98.0%

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

      7. distribute-neg-in 98.0%

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

      8. +-commutative 98.0%

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

      9. sub-neg 98.0%

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

      10. associate-*l* 98.0%

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

      11. sub-neg 98.0%

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

      12. +-commutative 98.0%

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

      13. distribute-neg-in 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in im around 0 8.0%

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

      1. log1p-expm1-u 97.6%

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

      2. associate-*l* 97.6%

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

   7. Applied egg-rr 97.6%

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

   8. Taylor expanded in re around 0 67.5%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot \color{blue}{im}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.2 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+110}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.2e+33)
   (* 0.5 (* (cos re) (* -2.0 im)))
   (if (<= im 1.4e+110)
     (* 0.5 (* im (+ -2.0 (pow re 2.0))))
     (* 0.5 (+ (* -2.0 im) (* -0.3333333333333333 (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.2e+33) {
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	} else if (im <= 1.4e+110) {
		tmp = 0.5 * (im * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = 0.5 * ((-2.0 * im) + (-0.3333333333333333 * pow(im, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.2d+33) then
        tmp = 0.5d0 * (cos(re) * ((-2.0d0) * im))
    else if (im <= 1.4d+110) then
        tmp = 0.5d0 * (im * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = 0.5d0 * (((-2.0d0) * im) + ((-0.3333333333333333d0) * (im ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.2e+33) {
		tmp = 0.5 * (Math.cos(re) * (-2.0 * im));
	} else if (im <= 1.4e+110) {
		tmp = 0.5 * (im * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * ((-2.0 * im) + (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.2e+33:
		tmp = 0.5 * (math.cos(re) * (-2.0 * im))
	elif im <= 1.4e+110:
		tmp = 0.5 * (im * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = 0.5 * ((-2.0 * im) + (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.2e+33)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im)));
	elseif (im <= 1.4e+110)
		tmp = Float64(0.5 * Float64(im * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(Float64(-2.0 * im) + Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.2e+33)
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	elseif (im <= 1.4e+110)
		tmp = 0.5 * (im * (-2.0 + (re ^ 2.0)));
	else
		tmp = 0.5 * ((-2.0 * im) + (-0.3333333333333333 * (im ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.2e+33], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+110], N[(0.5 * N[(im * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(-2.0 * im), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.20000000000000017e33

   1. Initial program 37.3%

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

      1. cos-neg 37.3%

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

      2. sub-neg 37.3%

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

      3. neg-sub0 37.3%

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

      4. remove-double-neg 37.3%

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

      5. remove-double-neg 37.3%

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

      6. sub0-neg 37.3%

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

      7. distribute-neg-in 37.3%

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

      8. +-commutative 37.3%

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

      9. sub-neg 37.3%

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

      10. associate-*l* 37.3%

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

      11. sub-neg 37.3%

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

      12. +-commutative 37.3%

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

      13. distribute-neg-in 37.3%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in im around 0 68.4%

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

   if 3.20000000000000017e33 < im < 1.39999999999999993e110

   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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   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 3.7%

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

   6. Taylor expanded in re around 0 28.9%

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

      1. *-commutative 28.9%

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

      2. distribute-lft-out 28.9%

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

   8. Simplified 28.9%

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

   if 1.39999999999999993e110 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 68.3%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.2 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+110}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+110}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.2e+33)
   (* 0.5 (* (cos re) (* -2.0 im)))
   (if (<= im 1.4e+110)
     (* 0.5 (* im (+ -2.0 (pow re 2.0))))
     (* (pow im 3.0) -0.16666666666666666))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.2e+33) {
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	} else if (im <= 1.4e+110) {
		tmp = 0.5 * (im * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 5.2d+33) then
        tmp = 0.5d0 * (cos(re) * ((-2.0d0) * im))
    else if (im <= 1.4d+110) then
        tmp = 0.5d0 * (im * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.2e+33) {
		tmp = 0.5 * (Math.cos(re) * (-2.0 * im));
	} else if (im <= 1.4e+110) {
		tmp = 0.5 * (im * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.2e+33:
		tmp = 0.5 * (math.cos(re) * (-2.0 * im))
	elif im <= 1.4e+110:
		tmp = 0.5 * (im * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.2e+33)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im)));
	elseif (im <= 1.4e+110)
		tmp = Float64(0.5 * Float64(im * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.2e+33)
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	elseif (im <= 1.4e+110)
		tmp = 0.5 * (im * (-2.0 + (re ^ 2.0)));
	else
		tmp = (im ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.2e+33], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+110], N[(0.5 * N[(im * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.1999999999999995e33

   1. Initial program 37.3%

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

      1. cos-neg 37.3%

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

      2. sub-neg 37.3%

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

      3. neg-sub0 37.3%

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

      4. remove-double-neg 37.3%

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

      5. remove-double-neg 37.3%

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

      6. sub0-neg 37.3%

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

      7. distribute-neg-in 37.3%

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

      8. +-commutative 37.3%

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

      9. sub-neg 37.3%

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

      10. associate-*l* 37.3%

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

      11. sub-neg 37.3%

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

      12. +-commutative 37.3%

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

      13. distribute-neg-in 37.3%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in im around 0 68.4%

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

   if 5.1999999999999995e33 < im < 1.39999999999999993e110

   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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   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 3.7%

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

   6. Taylor expanded in re around 0 28.9%

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

      1. *-commutative 28.9%

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

      2. distribute-lft-out 28.9%

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

   8. Simplified 28.9%

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

   if 1.39999999999999993e110 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left({im}^{3} \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around 0 68.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative 68.3%

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

   11. Simplified 68.3%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{+33}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+110}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.2 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.2e+67)
   (* 0.5 (* (cos re) (* -2.0 im)))
   (* (pow im 3.0) -0.16666666666666666)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.2e+67) {
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	} else {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.2d+67) then
        tmp = 0.5d0 * (cos(re) * ((-2.0d0) * im))
    else
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.2e+67) {
		tmp = 0.5 * (Math.cos(re) * (-2.0 * im));
	} else {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.2e+67:
		tmp = 0.5 * (math.cos(re) * (-2.0 * im))
	else:
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.2e+67)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im)));
	else
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.2e+67)
		tmp = 0.5 * (cos(re) * (-2.0 * im));
	else
		tmp = (im ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.2e+67], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.19999999999999983e67

   1. Initial program 40.3%

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

      1. cos-neg 40.3%

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

      2. sub-neg 40.3%

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

      3. neg-sub0 40.3%

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

      4. remove-double-neg 40.3%

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

      5. remove-double-neg 40.3%

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

      6. sub0-neg 40.3%

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

      7. distribute-neg-in 40.3%

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

      8. +-commutative 40.3%

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

      9. sub-neg 40.3%

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

      10. associate-*l* 40.3%

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

      11. sub-neg 40.3%

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

      12. +-commutative 40.3%

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

      13. distribute-neg-in 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in im around 0 65.3%

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

   if 3.19999999999999983e67 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   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 87.2%

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

   6. Taylor expanded in im around inf 87.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left({im}^{3} \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 87.2%

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

      2. associate-*l* 87.2%

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

   8. Simplified 87.2%

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

   9. Taylor expanded in re around 0 59.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative 59.0%

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

   11. Simplified 59.0%

       \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.2 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.5) (* 0.5 (* -2.0 im)) (* (pow im 3.0) -0.16666666666666666)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.5) {
		tmp = 0.5 * (-2.0 * im);
	} else {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d0) then
        tmp = 0.5d0 * ((-2.0d0) * im)
    else
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.5) {
		tmp = 0.5 * (-2.0 * im);
	} else {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.5:
		tmp = 0.5 * (-2.0 * im)
	else:
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.5)
		tmp = Float64(0.5 * Float64(-2.0 * im));
	else
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.5)
		tmp = 0.5 * (-2.0 * im);
	else
		tmp = (im ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5], N[(0.5 * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.5

   1. Initial program 33.9%

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

      1. cos-neg 33.9%

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

      2. sub-neg 33.9%

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

      3. neg-sub0 33.9%

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

      4. remove-double-neg 33.9%

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

      5. remove-double-neg 33.9%

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

      6. sub0-neg 33.9%

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

      7. distribute-neg-in 33.9%

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

      8. +-commutative 33.9%

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

      9. sub-neg 33.9%

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

      10. associate-*l* 33.9%

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

      11. sub-neg 33.9%

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

      12. +-commutative 33.9%

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

      13. distribute-neg-in 33.9%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in im around 0 71.9%

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

   6. Taylor expanded in re around 0 37.3%

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

   if 2.5 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   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 63.5%

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

   6. Taylor expanded in im around inf 63.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left({im}^{3} \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 63.5%

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

      2. associate-*l* 63.5%

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

   8. Simplified 63.5%

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

   9. Taylor expanded in re around 0 42.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative 42.9%

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

   11. Simplified 42.9%

       \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.4% 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 52.0%

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

   1. cos-neg 52.0%

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

   2. sub-neg 52.0%

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

   3. neg-sub0 52.0%

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

   4. remove-double-neg 52.0%

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

   5. remove-double-neg 52.0%

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

   6. sub0-neg 52.0%

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

   7. distribute-neg-in 52.0%

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

   8. +-commutative 52.0%

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

   9. sub-neg 52.0%

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

   10. associate-*l* 52.0%

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

   11. sub-neg 52.0%

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

   12. +-commutative 52.0%

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

   13. distribute-neg-in 52.0%

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

3. Simplified 52.0%

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

5. Taylor expanded in im around 0 53.7%

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

6. Taylor expanded in re around 0 28.1%

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

7. Final simplification 28.1%

   \[\leadsto 0.5 \cdot \left(-2 \cdot im\right) \]
8. Add Preprocessing

Developer target: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024017 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

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

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