math.sin on complex, imaginary part

Percentage Accurate: 54.0% → 98.9%

Time: 10.5s

Alternatives: 10

Speedup: 2.8×

• 48.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. sub-neg 48.7%

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

   2. neg-sub0 48.7%

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

   3. remove-double-neg 48.7%

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

   4. remove-double-neg 48.7%

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

   5. sub0-neg 48.7%

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

   6. distribute-neg-in 48.7%

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

   7. +-commutative 48.7%

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

   8. sub-neg 48.7%

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

   9. cos-neg 48.7%

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

   10. associate-*l* 48.7%

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

   11. distribute-rgt-neg-in 48.7%

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

   12. *-commutative 48.7%

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

3. Simplified 48.7%

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

5. Taylor expanded in im around 0 58.3%

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

   1. log1p-expm1-u 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*l* 99.3%

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

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 90.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 380:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{5} \cdot \left(\cos re \cdot -0.016666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 380.0)
   (* 0.5 (* (cos re) (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0)))))
   (if (<= im 4.5e+61)
     (* 0.5 (log1p (expm1 (* im -2.0))))
     (* 0.5 (* (pow im 5.0) (* (cos re) -0.016666666666666666))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 380.0) {
		tmp = 0.5 * (cos(re) * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0))));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	} else {
		tmp = 0.5 * (pow(im, 5.0) * (cos(re) * -0.016666666666666666));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 380.0) {
		tmp = 0.5 * (Math.cos(re) * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0))));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	} else {
		tmp = 0.5 * (Math.pow(im, 5.0) * (Math.cos(re) * -0.016666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 380.0:
		tmp = 0.5 * (math.cos(re) * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0))))
	elif im <= 4.5e+61:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	else:
		tmp = 0.5 * (math.pow(im, 5.0) * (math.cos(re) * -0.016666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 380.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0)))));
	elseif (im <= 4.5e+61)
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	else
		tmp = Float64(0.5 * Float64((im ^ 5.0) * Float64(cos(re) * -0.016666666666666666)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 380.0], 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], If[LessEqual[im, 4.5e+61], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[im, 5.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 380

   1. Initial program 36.5%

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

      1. sub-neg 36.5%

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

      2. neg-sub0 36.5%

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

      3. remove-double-neg 36.5%

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

      4. remove-double-neg 36.5%

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

      5. sub0-neg 36.5%

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

      6. distribute-neg-in 36.5%

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

      7. +-commutative 36.5%

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

      8. sub-neg 36.5%

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

      9. cos-neg 36.5%

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

      10. associate-*l* 36.5%

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

      11. distribute-rgt-neg-in 36.5%

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

      12. *-commutative 36.5%

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

   3. Simplified 36.5%

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

   5. Taylor expanded in im around 0 89.4%

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

   if 380 < im < 4.5e61

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.5%

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

      1. log1p-expm1-u 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 70.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{e^{-2 \cdot im} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 70.0%

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

   10. Simplified 70.0%

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

   if 4.5e61 < im

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 380:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{5} \cdot \left(\cos re \cdot -0.016666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{5} \cdot \left(\cos re \cdot -0.016666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 520.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 4.5e+61)
     (* 0.5 (log1p (expm1 (* im -2.0))))
     (* 0.5 (* (pow im 5.0) (* (cos re) -0.016666666666666666))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 520.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	} else {
		tmp = 0.5 * (pow(im, 5.0) * (cos(re) * -0.016666666666666666));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 520.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	} else {
		tmp = 0.5 * (Math.pow(im, 5.0) * (Math.cos(re) * -0.016666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 520.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 4.5e+61:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	else:
		tmp = 0.5 * (math.pow(im, 5.0) * (math.cos(re) * -0.016666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 520.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 4.5e+61)
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	else
		tmp = Float64(0.5 * Float64((im ^ 5.0) * Float64(cos(re) * -0.016666666666666666)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 520.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.5e+61], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[im, 5.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 520

   1. Initial program 36.5%

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

      1. sub-neg 36.5%

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

      2. neg-sub0 36.5%

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

      3. remove-double-neg 36.5%

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

      4. remove-double-neg 36.5%

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

      5. sub0-neg 36.5%

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

      6. distribute-neg-in 36.5%

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

      7. +-commutative 36.5%

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

      8. sub-neg 36.5%

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

      9. cos-neg 36.5%

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

      10. associate-*l* 36.5%

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

      11. distribute-rgt-neg-in 36.5%

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

      12. *-commutative 36.5%

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

   3. Simplified 36.5%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 520 < im < 4.5e61

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.5%

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

      1. log1p-expm1-u 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 70.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{e^{-2 \cdot im} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 70.0%

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

   10. Simplified 70.0%

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

   if 4.5e61 < im

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{5} \cdot \left(\cos re \cdot -0.016666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.005) (* 0.5 (* im (pow re 2.0))) (* 0.5 (* im -2.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.005) {
		tmp = 0.5 * (im * pow(re, 2.0));
	} 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) <= (-0.005d0)) then
        tmp = 0.5d0 * (im * (re ** 2.0d0))
    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) <= -0.005) {
		tmp = 0.5 * (im * Math.pow(re, 2.0));
	} else {
		tmp = 0.5 * (im * -2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= -0.005:
		tmp = 0.5 * (im * math.pow(re, 2.0))
	else:
		tmp = 0.5 * (im * -2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.005)
		tmp = Float64(0.5 * Float64(im * (re ^ 2.0)));
	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) <= -0.005)
		tmp = 0.5 * (im * (re ^ 2.0));
	else
		tmp = 0.5 * (im * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.005], N[(0.5 * N[(im * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0050000000000000001

   1. Initial program 44.4%

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

      1. sub-neg 44.4%

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

      2. neg-sub0 44.4%

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

      3. remove-double-neg 44.4%

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

      4. remove-double-neg 44.4%

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

      5. sub0-neg 44.4%

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

      6. distribute-neg-in 44.4%

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

      7. +-commutative 44.4%

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

      8. sub-neg 44.4%

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

      9. cos-neg 44.4%

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

      10. associate-*l* 44.4%

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

      11. distribute-rgt-neg-in 44.4%

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

      12. *-commutative 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in im around 0 62.4%

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

      1. log1p-expm1-u 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l* 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in re around 0 31.9%

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

      1. +-commutative 31.9%

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

      2. *-commutative 31.9%

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

      3. distribute-lft-out 31.9%

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

   10. Simplified 31.9%

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

   11. Taylor expanded in re around inf 31.9%

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

   if -0.0050000000000000001 < (cos.f64 re)

   1. Initial program 50.1%

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

      1. sub-neg 50.1%

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

      2. neg-sub0 50.1%

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

      3. remove-double-neg 50.1%

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

      4. remove-double-neg 50.1%

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

      5. sub0-neg 50.1%

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

      6. distribute-neg-in 50.1%

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

      7. +-commutative 50.1%

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

      8. sub-neg 50.1%

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

      9. cos-neg 50.1%

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

      10. associate-*l* 50.1%

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

      11. distribute-rgt-neg-in 50.1%

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

      12. *-commutative 50.1%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in im around 0 56.9%

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

   6. Taylor expanded in re around 0 41.6%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.005:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 440:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 440.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (* 0.5 (log1p (expm1 (* im -2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 440.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 440.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 440.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	else:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 440.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	else
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 440.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 440

   1. Initial program 36.5%

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

      1. sub-neg 36.5%

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

      2. neg-sub0 36.5%

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

      3. remove-double-neg 36.5%

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

      4. remove-double-neg 36.5%

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

      5. sub0-neg 36.5%

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

      6. distribute-neg-in 36.5%

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

      7. +-commutative 36.5%

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

      8. sub-neg 36.5%

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

      9. cos-neg 36.5%

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

      10. associate-*l* 36.5%

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

      11. distribute-rgt-neg-in 36.5%

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

      12. *-commutative 36.5%

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

   3. Simplified 36.5%

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

   5. Taylor expanded in im around 0 70.8%

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

   if 440 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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 5.2%

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

      1. log1p-expm1-u 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 81.6%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{e^{-2 \cdot im} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 81.6%

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

   10. Simplified 81.6%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 440:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2500000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2500000000.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 5.8e+59)
     (* 0.5 (* im (+ -2.0 (pow re 2.0))))
     (* 0.5 (+ (* im -2.0) (* (pow im 5.0) -0.016666666666666666))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2500000000.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 5.8e+59) {
		tmp = 0.5 * (im * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = 0.5 * ((im * -2.0) + (pow(im, 5.0) * -0.016666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2500000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else if (im <= 5.8d+59) then
        tmp = 0.5d0 * (im * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((im ** 5.0d0) * (-0.016666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2500000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 5.8e+59) {
		tmp = 0.5 * (im * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * ((im * -2.0) + (Math.pow(im, 5.0) * -0.016666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2500000000.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 5.8e+59:
		tmp = 0.5 * (im * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = 0.5 * ((im * -2.0) + (math.pow(im, 5.0) * -0.016666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2500000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 5.8e+59)
		tmp = Float64(0.5 * Float64(im * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64((im ^ 5.0) * -0.016666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2500000000.0)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	elseif (im <= 5.8e+59)
		tmp = 0.5 * (im * (-2.0 + (re ^ 2.0)));
	else
		tmp = 0.5 * ((im * -2.0) + ((im ^ 5.0) * -0.016666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2500000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.8e+59], N[(0.5 * N[(im * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.5e9

   1. Initial program 36.8%

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

      1. sub-neg 36.8%

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

      2. neg-sub0 36.8%

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

      3. remove-double-neg 36.8%

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

      4. remove-double-neg 36.8%

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

      5. sub0-neg 36.8%

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

      6. distribute-neg-in 36.8%

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

      7. +-commutative 36.8%

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

      8. sub-neg 36.8%

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

      9. cos-neg 36.8%

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

      10. associate-*l* 36.8%

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

      11. distribute-rgt-neg-in 36.8%

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

      12. *-commutative 36.8%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in im around 0 70.5%

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

   if 2.5e9 < im < 5.79999999999999981e59

   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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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 3.4%

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

   6. Taylor expanded in re around 0 30.3%

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

      1. *-commutative 30.3%

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

      2. distribute-lft-out 30.3%

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

   8. Simplified 30.3%

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

   if 5.79999999999999981e59 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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 95.7%

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

   6. Taylor expanded in re around 0 81.1%

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

   7. Taylor expanded in im around inf 81.1%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2500000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 580000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{5} \cdot -0.016666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 580000000.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 1.55e+60)
     (* 0.5 (* im (+ -2.0 (pow re 2.0))))
     (* 0.5 (* (pow im 5.0) -0.016666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 580000000.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 1.55e+60) {
		tmp = 0.5 * (im * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = 0.5 * (pow(im, 5.0) * -0.016666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 580000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else if (im <= 1.55d+60) then
        tmp = 0.5d0 * (im * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = 0.5d0 * ((im ** 5.0d0) * (-0.016666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 580000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 1.55e+60) {
		tmp = 0.5 * (im * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * (Math.pow(im, 5.0) * -0.016666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 580000000.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 1.55e+60:
		tmp = 0.5 * (im * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = 0.5 * (math.pow(im, 5.0) * -0.016666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 580000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 1.55e+60)
		tmp = Float64(0.5 * Float64(im * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64((im ^ 5.0) * -0.016666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 580000000.0)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	elseif (im <= 1.55e+60)
		tmp = 0.5 * (im * (-2.0 + (re ^ 2.0)));
	else
		tmp = 0.5 * ((im ^ 5.0) * -0.016666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 580000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.55e+60], N[(0.5 * N[(im * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.8e8

   1. Initial program 36.8%

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

      1. sub-neg 36.8%

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

      2. neg-sub0 36.8%

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

      3. remove-double-neg 36.8%

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

      4. remove-double-neg 36.8%

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

      5. sub0-neg 36.8%

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

      6. distribute-neg-in 36.8%

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

      7. +-commutative 36.8%

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

      8. sub-neg 36.8%

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

      9. cos-neg 36.8%

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

      10. associate-*l* 36.8%

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

      11. distribute-rgt-neg-in 36.8%

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

      12. *-commutative 36.8%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in im around 0 70.5%

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

   if 5.8e8 < im < 1.55e60

   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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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 3.4%

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

   6. Taylor expanded in re around 0 30.3%

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

      1. *-commutative 30.3%

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

      2. distribute-lft-out 30.3%

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

   8. Simplified 30.3%

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

   if 1.55e60 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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 95.7%

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

   6. Taylor expanded in im around inf 95.7%

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

      1. *-commutative 95.7%

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

      2. associate-*l* 95.7%

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

   8. Simplified 95.7%

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

   9. Taylor expanded in re around 0 81.1%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 580000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{5} \cdot -0.016666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 550000000:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{5} \cdot -0.016666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 550000000.0)
   (* 0.5 (* im -2.0))
   (if (<= im 2.5e+58)
     (* 0.5 (* im (pow re 2.0)))
     (* 0.5 (* (pow im 5.0) -0.016666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 550000000.0) {
		tmp = 0.5 * (im * -2.0);
	} else if (im <= 2.5e+58) {
		tmp = 0.5 * (im * pow(re, 2.0));
	} else {
		tmp = 0.5 * (pow(im, 5.0) * -0.016666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 550000000.0d0) then
        tmp = 0.5d0 * (im * (-2.0d0))
    else if (im <= 2.5d+58) then
        tmp = 0.5d0 * (im * (re ** 2.0d0))
    else
        tmp = 0.5d0 * ((im ** 5.0d0) * (-0.016666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 550000000.0) {
		tmp = 0.5 * (im * -2.0);
	} else if (im <= 2.5e+58) {
		tmp = 0.5 * (im * Math.pow(re, 2.0));
	} else {
		tmp = 0.5 * (Math.pow(im, 5.0) * -0.016666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 550000000.0:
		tmp = 0.5 * (im * -2.0)
	elif im <= 2.5e+58:
		tmp = 0.5 * (im * math.pow(re, 2.0))
	else:
		tmp = 0.5 * (math.pow(im, 5.0) * -0.016666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 550000000.0)
		tmp = Float64(0.5 * Float64(im * -2.0));
	elseif (im <= 2.5e+58)
		tmp = Float64(0.5 * Float64(im * (re ^ 2.0)));
	else
		tmp = Float64(0.5 * Float64((im ^ 5.0) * -0.016666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 550000000.0)
		tmp = 0.5 * (im * -2.0);
	elseif (im <= 2.5e+58)
		tmp = 0.5 * (im * (re ^ 2.0));
	else
		tmp = 0.5 * ((im ^ 5.0) * -0.016666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 550000000.0], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+58], N[(0.5 * N[(im * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.5e8

   1. Initial program 36.8%

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

      1. sub-neg 36.8%

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

      2. neg-sub0 36.8%

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

      3. remove-double-neg 36.8%

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

      4. remove-double-neg 36.8%

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

      5. sub0-neg 36.8%

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

      6. distribute-neg-in 36.8%

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

      7. +-commutative 36.8%

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

      8. sub-neg 36.8%

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

      9. cos-neg 36.8%

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

      10. associate-*l* 36.8%

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

      11. distribute-rgt-neg-in 36.8%

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

      12. *-commutative 36.8%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in im around 0 70.5%

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

   6. Taylor expanded in re around 0 38.1%

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

   if 5.5e8 < im < 2.49999999999999993e58

   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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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 3.4%

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

      1. log1p-expm1-u 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 30.3%

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

      1. +-commutative 30.3%

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

      2. *-commutative 30.3%

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

      3. distribute-lft-out 30.3%

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

   10. Simplified 30.3%

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

   11. Taylor expanded in re around inf 29.8%

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

   if 2.49999999999999993e58 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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 95.7%

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

   6. Taylor expanded in im around inf 95.7%

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

      1. *-commutative 95.7%

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

      2. associate-*l* 95.7%

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

   8. Simplified 95.7%

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

   9. Taylor expanded in re around 0 81.1%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550000000:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{5} \cdot -0.016666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1150000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{5} \cdot -0.016666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1150000000.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 6.2e+59)
     (* 0.5 (* im (pow re 2.0)))
     (* 0.5 (* (pow im 5.0) -0.016666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1150000000.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 6.2e+59) {
		tmp = 0.5 * (im * pow(re, 2.0));
	} else {
		tmp = 0.5 * (pow(im, 5.0) * -0.016666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1150000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else if (im <= 6.2d+59) then
        tmp = 0.5d0 * (im * (re ** 2.0d0))
    else
        tmp = 0.5d0 * ((im ** 5.0d0) * (-0.016666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1150000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 6.2e+59) {
		tmp = 0.5 * (im * Math.pow(re, 2.0));
	} else {
		tmp = 0.5 * (Math.pow(im, 5.0) * -0.016666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1150000000.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 6.2e+59:
		tmp = 0.5 * (im * math.pow(re, 2.0))
	else:
		tmp = 0.5 * (math.pow(im, 5.0) * -0.016666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1150000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 6.2e+59)
		tmp = Float64(0.5 * Float64(im * (re ^ 2.0)));
	else
		tmp = Float64(0.5 * Float64((im ^ 5.0) * -0.016666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1150000000.0)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	elseif (im <= 6.2e+59)
		tmp = 0.5 * (im * (re ^ 2.0));
	else
		tmp = 0.5 * ((im ^ 5.0) * -0.016666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1150000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.2e+59], N[(0.5 * N[(im * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.15e9

   1. Initial program 36.8%

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

      1. sub-neg 36.8%

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

      2. neg-sub0 36.8%

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

      3. remove-double-neg 36.8%

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

      4. remove-double-neg 36.8%

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

      5. sub0-neg 36.8%

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

      6. distribute-neg-in 36.8%

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

      7. +-commutative 36.8%

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

      8. sub-neg 36.8%

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

      9. cos-neg 36.8%

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

      10. associate-*l* 36.8%

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

      11. distribute-rgt-neg-in 36.8%

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

      12. *-commutative 36.8%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in im around 0 70.5%

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

   if 1.15e9 < im < 6.20000000000000029e59

   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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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 3.4%

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

      1. log1p-expm1-u 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 30.3%

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

      1. +-commutative 30.3%

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

      2. *-commutative 30.3%

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

      3. distribute-lft-out 30.3%

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

   10. Simplified 30.3%

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

   11. Taylor expanded in re around inf 29.8%

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

   if 6.20000000000000029e59 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      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. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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 95.7%

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

   6. Taylor expanded in im around inf 95.7%

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

      1. *-commutative 95.7%

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

      2. associate-*l* 95.7%

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

   8. Simplified 95.7%

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

   9. Taylor expanded in re around 0 81.1%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1150000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{5} \cdot -0.016666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.4% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. sub-neg 48.7%

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

   2. neg-sub0 48.7%

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

   3. remove-double-neg 48.7%

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

   4. remove-double-neg 48.7%

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

   5. sub0-neg 48.7%

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

   6. distribute-neg-in 48.7%

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

   7. +-commutative 48.7%

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

   8. sub-neg 48.7%

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

   9. cos-neg 48.7%

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

   10. associate-*l* 48.7%

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

   11. distribute-rgt-neg-in 48.7%

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

   12. *-commutative 48.7%

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

3. Simplified 48.7%

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

5. Taylor expanded in im around 0 58.3%

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

6. Taylor expanded in re around 0 31.8%

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

7. Final simplification 31.8%

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

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024013 
(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))))