math.sin on complex, imaginary part

Percentage Accurate: 54.4% → 99.0%

Time: 10.9s

Alternatives: 12

Speedup: 2.8×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. cos-neg 54.6%

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

   2. sub-neg 54.6%

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

   3. neg-sub0 54.6%

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

   4. remove-double-neg 54.6%

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

   5. remove-double-neg 54.6%

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

   6. sub0-neg 54.6%

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

   7. distribute-neg-in 54.6%

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

   8. +-commutative 54.6%

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

   9. sub-neg 54.6%

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

   10. associate-*l* 54.6%

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

   11. sub-neg 54.6%

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

   12. +-commutative 54.6%

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

   13. distribute-neg-in 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in im around 0 52.1%

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

   1. log1p-expm1-u 99.8%

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

   2. associate-*r* 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-*r* 99.8%

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

   5. metadata-eval 99.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \left(\color{blue}{-1} \cdot im\right)\right)\right) \]

7. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \left(-1 \cdot im\right)\right)\right)} \]

8. Final simplification 99.8%

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

Alternative 2: 69.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3800000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+97}:\\ \;\;\;\;-0.041666666666666664 \cdot \left(im \cdot {re}^{4}\right) - im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3800000.0)
   (* (cos re) (- im))
   (if (<= im 7e+97)
     (- (* -0.041666666666666664 (* im (pow re 4.0))) im)
     (* 0.5 (* (cos re) (* -0.3333333333333333 (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3800000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 7e+97) {
		tmp = (-0.041666666666666664 * (im * pow(re, 4.0))) - im;
	} else {
		tmp = 0.5 * (cos(re) * (-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 <= 3800000.0d0) then
        tmp = cos(re) * -im
    else if (im <= 7d+97) then
        tmp = ((-0.041666666666666664d0) * (im * (re ** 4.0d0))) - im
    else
        tmp = 0.5d0 * (cos(re) * ((-0.3333333333333333d0) * (im ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3800000.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 7e+97) {
		tmp = (-0.041666666666666664 * (im * Math.pow(re, 4.0))) - im;
	} else {
		tmp = 0.5 * (Math.cos(re) * (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3800000.0:
		tmp = math.cos(re) * -im
	elif im <= 7e+97:
		tmp = (-0.041666666666666664 * (im * math.pow(re, 4.0))) - im
	else:
		tmp = 0.5 * (math.cos(re) * (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3800000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 7e+97)
		tmp = Float64(Float64(-0.041666666666666664 * Float64(im * (re ^ 4.0))) - im);
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3800000.0)
		tmp = cos(re) * -im;
	elseif (im <= 7e+97)
		tmp = (-0.041666666666666664 * (im * (re ^ 4.0))) - im;
	else
		tmp = 0.5 * (cos(re) * (-0.3333333333333333 * (im ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3800000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 7e+97], N[(N[(-0.041666666666666664 * N[(im * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.8e6

   1. Initial program 38.8%

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

      1. cos-neg 38.8%

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

      2. sub-neg 38.8%

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

      3. neg-sub0 38.8%

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

      4. remove-double-neg 38.8%

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

      5. remove-double-neg 38.8%

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

      6. sub0-neg 38.8%

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

      7. distribute-neg-in 38.8%

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

      8. +-commutative 38.8%

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

      9. sub-neg 38.8%

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

      10. associate-*l* 38.8%

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

      11. sub-neg 38.8%

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

      12. +-commutative 38.8%

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

      13. distribute-neg-in 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in im around 0 67.8%

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

   6. Taylor expanded in im around 0 67.3%

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

      1. associate-*r* 67.3%

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

      2. *-commutative 67.3%

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

      3. mul-1-neg 67.3%

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

   8. Simplified 67.3%

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

   if 3.8e6 < im < 7.0000000000000001e97

   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 3.6%

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

   6. Taylor expanded in re around 0 2.6%

      \[\leadsto \color{blue}{-1 \cdot im + \left(-0.041666666666666664 \cdot \left(im \cdot {re}^{4}\right) + 0.5 \cdot \left(im \cdot {re}^{2}\right)\right)} \]

   7. Taylor expanded in re around inf 21.3%

      \[\leadsto -1 \cdot im + \color{blue}{-0.041666666666666664 \cdot \left(im \cdot {re}^{4}\right)} \]

   if 7.0000000000000001e97 < 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 96.4%

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

   6. Taylor expanded in im around inf 96.4%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3800000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+97}:\\ \;\;\;\;-0.041666666666666664 \cdot \left(im \cdot {re}^{4}\right) - im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (* (cos re) (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0))))))

C

double code(double re, double im) {
	return 0.5 * (cos(re) * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (cos(re) * ((im * (-2.0d0)) + ((-0.3333333333333333d0) * (im ** 3.0d0))))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (Math.cos(re) * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0))));
}

Python

def code(re, im):
	return 0.5 * (math.cos(re) * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0))))

Julia

function code(re, im)
	return Float64(0.5 * Float64(cos(re) * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0)))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (cos(re) * ((im * -2.0) + (-0.3333333333333333 * (im ^ 3.0))));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. cos-neg 54.6%

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

   2. sub-neg 54.6%

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

   3. neg-sub0 54.6%

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

   4. remove-double-neg 54.6%

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

   5. remove-double-neg 54.6%

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

   6. sub0-neg 54.6%

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

   7. distribute-neg-in 54.6%

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

   8. +-commutative 54.6%

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

   9. sub-neg 54.6%

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

   10. associate-*l* 54.6%

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

   11. sub-neg 54.6%

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

   12. +-commutative 54.6%

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

   13. distribute-neg-in 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in im around 0 84.8%

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

6. Final simplification 84.8%

   \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\right) \]
7. Add Preprocessing

Alternative 4: 64.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.5 \cdot 10^{+42}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.5e+42)
   (* (cos re) (- im))
   (if (<= im 2e+105)
     (log1p (expm1 im))
     (* 0.5 (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.5e+42) {
		tmp = cos(re) * -im;
	} else if (im <= 2e+105) {
		tmp = log1p(expm1(im));
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.5e+42) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 2e+105) {
		tmp = Math.log1p(Math.expm1(im));
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.5e+42:
		tmp = math.cos(re) * -im
	elif im <= 2e+105:
		tmp = math.log1p(math.expm1(im))
	else:
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.5e+42)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 2e+105)
		tmp = log1p(expm1(im));
	else
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.5e+42], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 2e+105], N[Log[1 + N[(Exp[im] - 1), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.50000000000000052e42

   1. Initial program 39.7%

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

      1. cos-neg 39.7%

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

      2. sub-neg 39.7%

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

      3. neg-sub0 39.7%

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

      4. remove-double-neg 39.7%

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

      5. remove-double-neg 39.7%

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

      6. sub0-neg 39.7%

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

      7. distribute-neg-in 39.7%

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

      8. +-commutative 39.7%

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

      9. sub-neg 39.7%

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

      10. associate-*l* 39.7%

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

      11. sub-neg 39.7%

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

      12. +-commutative 39.7%

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

      13. distribute-neg-in 39.7%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in im around 0 66.8%

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

   6. Taylor expanded in im around 0 66.3%

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

      1. associate-*r* 66.3%

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

      2. *-commutative 66.3%

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

      3. mul-1-neg 66.3%

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

   8. Simplified 66.3%

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

   if 6.50000000000000052e42 < im < 1.9999999999999999e105

   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 3.8%

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

   6. Taylor expanded in re around 0 2.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 1.3%

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

      3. associate-*r* 1.3%

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

      4. associate-*r* 1.3%

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

      5. swap-sqr 1.3%

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

      6. metadata-eval 1.3%

         \[\leadsto \sqrt{\left(\color{blue}{-1} \cdot \left(0.5 \cdot -2\right)\right) \cdot \left(im \cdot im\right)} \]

      7. metadata-eval 1.3%

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

      8. metadata-eval 1.3%

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

      9. *-un-lft-identity 1.3%

         \[\leadsto \sqrt{\color{blue}{im \cdot im}} \]

      10. sqrt-unprod 1.3%

          \[\leadsto \color{blue}{\sqrt{im} \cdot \sqrt{im}} \]

      11. add-sqr-sqrt 1.3%

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

      12. log1p-expm1-u 25.0%

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

   8. Applied egg-rr 25.0%

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

   if 1.9999999999999999e105 < 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 80.9%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.5 \cdot 10^{+42}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.8 \cdot 10^{+40}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(-1 + 0.5 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.8e+40)
   (* (cos re) (- im))
   (if (<= im 2e+105)
     (* im (+ -1.0 (* 0.5 (pow re 2.0))))
     (* 0.5 (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.8e+40) {
		tmp = cos(re) * -im;
	} else if (im <= 2e+105) {
		tmp = im * (-1.0 + (0.5 * pow(re, 2.0)));
	} else {
		tmp = 0.5 * ((im * -2.0) + (-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 <= 3.8d+40) then
        tmp = cos(re) * -im
    else if (im <= 2d+105) then
        tmp = im * ((-1.0d0) + (0.5d0 * (re ** 2.0d0)))
    else
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((-0.3333333333333333d0) * (im ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.8e+40) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 2e+105) {
		tmp = im * (-1.0 + (0.5 * Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.8e+40:
		tmp = math.cos(re) * -im
	elif im <= 2e+105:
		tmp = im * (-1.0 + (0.5 * math.pow(re, 2.0)))
	else:
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.8e+40)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 2e+105)
		tmp = Float64(im * Float64(-1.0 + Float64(0.5 * (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.8e+40)
		tmp = cos(re) * -im;
	elseif (im <= 2e+105)
		tmp = im * (-1.0 + (0.5 * (re ^ 2.0)));
	else
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * (im ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.8e+40], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 2e+105], N[(im * N[(-1.0 + N[(0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.80000000000000004e40

   1. Initial program 39.7%

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

      1. cos-neg 39.7%

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

      2. sub-neg 39.7%

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

      3. neg-sub0 39.7%

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

      4. remove-double-neg 39.7%

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

      5. remove-double-neg 39.7%

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

      6. sub0-neg 39.7%

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

      7. distribute-neg-in 39.7%

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

      8. +-commutative 39.7%

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

      9. sub-neg 39.7%

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

      10. associate-*l* 39.7%

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

      11. sub-neg 39.7%

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

      12. +-commutative 39.7%

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

      13. distribute-neg-in 39.7%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in im around 0 66.8%

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

   6. Taylor expanded in im around 0 66.3%

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

      1. associate-*r* 66.3%

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

      2. *-commutative 66.3%

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

      3. mul-1-neg 66.3%

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

   8. Simplified 66.3%

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

   if 3.80000000000000004e40 < im < 1.9999999999999999e105

   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 3.8%

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

   6. Taylor expanded in re around 0 27.1%

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

      1. *-commutative 27.1%

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

      2. *-commutative 27.1%

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

      3. associate-*l* 27.1%

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

      4. distribute-lft-out 27.1%

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

   8. Simplified 27.1%

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

   if 1.9999999999999999e105 < 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 80.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.8 \cdot 10^{+40}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(-1 + 0.5 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 480000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+97}:\\ \;\;\;\;-0.041666666666666664 \cdot \left(im \cdot {re}^{4}\right) - im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 480000.0)
   (* (cos re) (- im))
   (if (<= im 8e+97)
     (- (* -0.041666666666666664 (* im (pow re 4.0))) im)
     (* 0.5 (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 480000.0) {
		tmp = cos(re) * -im;
	} else if (im <= 8e+97) {
		tmp = (-0.041666666666666664 * (im * pow(re, 4.0))) - im;
	} else {
		tmp = 0.5 * ((im * -2.0) + (-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 <= 480000.0d0) then
        tmp = cos(re) * -im
    else if (im <= 8d+97) then
        tmp = ((-0.041666666666666664d0) * (im * (re ** 4.0d0))) - im
    else
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((-0.3333333333333333d0) * (im ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 480000.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 8e+97) {
		tmp = (-0.041666666666666664 * (im * Math.pow(re, 4.0))) - im;
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 480000.0:
		tmp = math.cos(re) * -im
	elif im <= 8e+97:
		tmp = (-0.041666666666666664 * (im * math.pow(re, 4.0))) - im
	else:
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 480000.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 8e+97)
		tmp = Float64(Float64(-0.041666666666666664 * Float64(im * (re ^ 4.0))) - im);
	else
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 480000.0)
		tmp = cos(re) * -im;
	elseif (im <= 8e+97)
		tmp = (-0.041666666666666664 * (im * (re ^ 4.0))) - im;
	else
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * (im ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 480000.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 8e+97], N[(N[(-0.041666666666666664 * N[(im * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.8e5

   1. Initial program 38.8%

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

      1. cos-neg 38.8%

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

      2. sub-neg 38.8%

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

      3. neg-sub0 38.8%

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

      4. remove-double-neg 38.8%

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

      5. remove-double-neg 38.8%

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

      6. sub0-neg 38.8%

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

      7. distribute-neg-in 38.8%

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

      8. +-commutative 38.8%

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

      9. sub-neg 38.8%

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

      10. associate-*l* 38.8%

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

      11. sub-neg 38.8%

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

      12. +-commutative 38.8%

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

      13. distribute-neg-in 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in im around 0 67.8%

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

   6. Taylor expanded in im around 0 67.3%

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

      1. associate-*r* 67.3%

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

      2. *-commutative 67.3%

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

      3. mul-1-neg 67.3%

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

   8. Simplified 67.3%

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

   if 4.8e5 < im < 8.0000000000000006e97

   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 3.6%

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

   6. Taylor expanded in re around 0 2.6%

      \[\leadsto \color{blue}{-1 \cdot im + \left(-0.041666666666666664 \cdot \left(im \cdot {re}^{4}\right) + 0.5 \cdot \left(im \cdot {re}^{2}\right)\right)} \]

   7. Taylor expanded in re around inf 21.3%

      \[\leadsto -1 \cdot im + \color{blue}{-0.041666666666666664 \cdot \left(im \cdot {re}^{4}\right)} \]

   if 8.0000000000000006e97 < 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 96.4%

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

   6. Taylor expanded in re around 0 76.4%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 480000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+97}:\\ \;\;\;\;-0.041666666666666664 \cdot \left(im \cdot {re}^{4}\right) - im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{+40}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(-1 + 0.5 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.2e+40)
   (* (cos re) (- im))
   (if (<= im 2e+105)
     (* im (+ -1.0 (* 0.5 (pow re 2.0))))
     (* (pow im 3.0) -0.16666666666666666))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.2e+40) {
		tmp = cos(re) * -im;
	} else if (im <= 2e+105) {
		tmp = im * (-1.0 + (0.5 * 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+40) then
        tmp = cos(re) * -im
    else if (im <= 2d+105) then
        tmp = im * ((-1.0d0) + (0.5d0 * (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+40) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 2e+105) {
		tmp = im * (-1.0 + (0.5 * 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+40:
		tmp = math.cos(re) * -im
	elif im <= 2e+105:
		tmp = im * (-1.0 + (0.5 * 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+40)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 2e+105)
		tmp = Float64(im * Float64(-1.0 + Float64(0.5 * (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+40)
		tmp = cos(re) * -im;
	elseif (im <= 2e+105)
		tmp = im * (-1.0 + (0.5 * (re ^ 2.0)));
	else
		tmp = (im ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.2e+40], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 2e+105], N[(im * N[(-1.0 + N[(0.5 * 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 < 5.2000000000000001e40

   1. Initial program 39.7%

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

      1. cos-neg 39.7%

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

      2. sub-neg 39.7%

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

      3. neg-sub0 39.7%

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

      4. remove-double-neg 39.7%

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

      5. remove-double-neg 39.7%

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

      6. sub0-neg 39.7%

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

      7. distribute-neg-in 39.7%

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

      8. +-commutative 39.7%

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

      9. sub-neg 39.7%

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

      10. associate-*l* 39.7%

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

      11. sub-neg 39.7%

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

      12. +-commutative 39.7%

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

      13. distribute-neg-in 39.7%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in im around 0 66.8%

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

   6. Taylor expanded in im around 0 66.3%

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

      1. associate-*r* 66.3%

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

      2. *-commutative 66.3%

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

      3. mul-1-neg 66.3%

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

   8. Simplified 66.3%

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

   if 5.2000000000000001e40 < im < 1.9999999999999999e105

   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 3.8%

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

   6. Taylor expanded in re around 0 27.1%

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

      1. *-commutative 27.1%

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

      2. *-commutative 27.1%

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

      3. associate-*l* 27.1%

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

      4. distribute-lft-out 27.1%

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

   8. Simplified 27.1%

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

   if 1.9999999999999999e105 < 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%

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

   7. Taylor expanded in re around 0 80.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2 \cdot 10^{+40}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+105}:\\ \;\;\;\;im \cdot \left(-1 + 0.5 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -4e-310) im (* 0.5 (* im -2.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -4e-310) {
		tmp = im;
	} else {
		tmp = 0.5 * (im * -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= (-4d-310)) then
        tmp = im
    else
        tmp = 0.5d0 * (im * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= -4e-310) {
		tmp = im;
	} else {
		tmp = 0.5 * (im * -2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= -4e-310:
		tmp = im
	else:
		tmp = 0.5 * (im * -2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -4e-310)
		tmp = im;
	else
		tmp = Float64(0.5 * Float64(im * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= -4e-310)
		tmp = im;
	else
		tmp = 0.5 * (im * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -4e-310], im, N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -3.999999999999988e-310

   1. Initial program 53.6%

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

      1. cos-neg 53.6%

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

      2. sub-neg 53.6%

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

      3. neg-sub0 53.6%

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

      4. remove-double-neg 53.6%

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

      5. remove-double-neg 53.6%

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

      6. sub0-neg 53.6%

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

      7. distribute-neg-in 53.6%

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

      8. +-commutative 53.6%

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

      9. sub-neg 53.6%

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

      10. associate-*l* 53.6%

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

      11. sub-neg 53.6%

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

      12. +-commutative 53.6%

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

      13. distribute-neg-in 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in im around 0 53.8%

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

   6. Taylor expanded in re around 0 1.9%

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

      1. add-sqr-sqrt 0.9%

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

      2. sqrt-unprod 15.9%

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

      3. associate-*r* 15.9%

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

      4. associate-*r* 15.9%

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

      5. swap-sqr 15.9%

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

      6. metadata-eval 15.9%

         \[\leadsto \sqrt{\left(\color{blue}{-1} \cdot \left(0.5 \cdot -2\right)\right) \cdot \left(im \cdot im\right)} \]

      7. metadata-eval 15.9%

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

      8. metadata-eval 15.9%

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

      9. *-un-lft-identity 15.9%

         \[\leadsto \sqrt{\color{blue}{im \cdot im}} \]

      10. sqrt-unprod 7.0%

          \[\leadsto \color{blue}{\sqrt{im} \cdot \sqrt{im}} \]

      11. add-sqr-sqrt 13.4%

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

      12. expm1-log1p-u 11.7%

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

      13. expm1-udef 4.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(im\right)} - 1} \]

   8. Applied egg-rr 4.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(im\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 11.7%

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

      2. expm1-log1p 13.4%

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

   10. Simplified 13.4%

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

   if -3.999999999999988e-310 < (cos.f64 re)

   1. Initial program 54.9%

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

      1. cos-neg 54.9%

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

      2. sub-neg 54.9%

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

      3. neg-sub0 54.9%

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

      4. remove-double-neg 54.9%

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

      5. remove-double-neg 54.9%

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

      6. sub0-neg 54.9%

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

      7. distribute-neg-in 54.9%

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

      8. +-commutative 54.9%

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

      9. sub-neg 54.9%

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

      10. associate-*l* 54.9%

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

      11. sub-neg 54.9%

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

      12. +-commutative 54.9%

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

      13. distribute-neg-in 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in im around 0 51.6%

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

   6. Taylor expanded in re around 0 43.4%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

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

C

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

Python

def code(re, im):
	tmp = 0
	if im <= 2.5:
		tmp = -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(-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 = -im;
	else
		tmp = (im ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5], (-im), 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 38.8%

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

      1. cos-neg 38.8%

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

      2. sub-neg 38.8%

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

      3. neg-sub0 38.8%

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

      4. remove-double-neg 38.8%

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

      5. remove-double-neg 38.8%

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

      6. sub0-neg 38.8%

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

      7. distribute-neg-in 38.8%

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

      8. +-commutative 38.8%

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

      9. sub-neg 38.8%

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

      10. associate-*l* 38.8%

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

      11. sub-neg 38.8%

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

      12. +-commutative 38.8%

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

      13. distribute-neg-in 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in im around 0 67.8%

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

   6. Taylor expanded in re around 0 43.3%

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

      1. mul-1-neg 43.3%

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

   8. Simplified 43.3%

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

   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 74.4%

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

   6. Taylor expanded in im around inf 74.4%

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

   7. Taylor expanded in re around 0 59.0%

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

4. Final simplification 47.4%

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

Alternative 10: 64.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8 \cdot 10^{+51}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8e+51) (* (cos re) (- im)) (* (pow im 3.0) -0.16666666666666666)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8e+51) {
		tmp = cos(re) * -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 <= 8d+51) then
        tmp = cos(re) * -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 <= 8e+51) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8e+51:
		tmp = math.cos(re) * -im
	else:
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8e+51)
		tmp = Float64(cos(re) * Float64(-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 <= 8e+51)
		tmp = cos(re) * -im;
	else
		tmp = (im ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8e+51], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8e51

   1. Initial program 40.7%

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

      1. cos-neg 40.7%

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

      2. sub-neg 40.7%

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

      3. neg-sub0 40.7%

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

      4. remove-double-neg 40.7%

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

      5. remove-double-neg 40.7%

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

      6. sub0-neg 40.7%

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

      7. distribute-neg-in 40.7%

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

      8. +-commutative 40.7%

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

      9. sub-neg 40.7%

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

      10. associate-*l* 40.7%

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

      11. sub-neg 40.7%

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

      12. +-commutative 40.7%

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

      13. distribute-neg-in 40.7%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in im around 0 65.8%

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

   6. Taylor expanded in im around 0 65.4%

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

      1. associate-*r* 65.4%

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

      2. *-commutative 65.4%

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

      3. mul-1-neg 65.4%

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

   8. Simplified 65.4%

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

   if 8e51 < 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 81.4%

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

   6. Taylor expanded in im around inf 81.4%

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

   7. Taylor expanded in re around 0 64.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8 \cdot 10^{+51}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.5% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. cos-neg 54.6%

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

   2. sub-neg 54.6%

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

   3. neg-sub0 54.6%

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

   4. remove-double-neg 54.6%

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

   5. remove-double-neg 54.6%

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

   6. sub0-neg 54.6%

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

   7. distribute-neg-in 54.6%

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

   8. +-commutative 54.6%

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

   9. sub-neg 54.6%

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

   10. associate-*l* 54.6%

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

   11. sub-neg 54.6%

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

   12. +-commutative 54.6%

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

   13. distribute-neg-in 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in im around 0 52.1%

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

6. Taylor expanded in re around 0 33.4%

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

   1. mul-1-neg 33.4%

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

8. Simplified 33.4%

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

9. Final simplification 33.4%

   \[\leadsto -im \]
10. Add Preprocessing

Alternative 12: 5.0% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 54.6%

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

   1. cos-neg 54.6%

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

   2. sub-neg 54.6%

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

   3. neg-sub0 54.6%

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

   4. remove-double-neg 54.6%

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

   5. remove-double-neg 54.6%

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

   6. sub0-neg 54.6%

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

   7. distribute-neg-in 54.6%

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

   8. +-commutative 54.6%

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

   9. sub-neg 54.6%

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

   10. associate-*l* 54.6%

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

   11. sub-neg 54.6%

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

   12. +-commutative 54.6%

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

   13. distribute-neg-in 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in im around 0 52.1%

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

6. Taylor expanded in re around 0 33.7%

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

   1. add-sqr-sqrt 16.8%

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

   2. sqrt-unprod 22.1%

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

   3. associate-*r* 22.1%

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

   4. associate-*r* 22.1%

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

   5. swap-sqr 22.1%

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

   6. metadata-eval 22.1%

      \[\leadsto \sqrt{\left(\color{blue}{-1} \cdot \left(0.5 \cdot -2\right)\right) \cdot \left(im \cdot im\right)} \]

   7. metadata-eval 22.1%

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

   8. metadata-eval 22.1%

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

   9. *-un-lft-identity 22.1%

      \[\leadsto \sqrt{\color{blue}{im \cdot im}} \]

   10. sqrt-unprod 2.4%

       \[\leadsto \color{blue}{\sqrt{im} \cdot \sqrt{im}} \]

   11. add-sqr-sqrt 4.7%

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

   12. expm1-log1p-u 4.2%

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

   13. expm1-udef 3.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(im\right)} - 1} \]

8. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(im\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 4.2%

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

   2. expm1-log1p 4.7%

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

10. Simplified 4.7%

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

11. Final simplification 4.7%

    \[\leadsto im \]
12. 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 2024027 
(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))))