math.sin on complex, imaginary part

Percentage Accurate: 54.6% → 99.0%

Time: 8.1s

Alternatives: 10

Speedup: 2.9×

• 49.9% 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: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

   1. sub-neg 54.5%

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

   2. neg-sub0 54.5%

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

   3. remove-double-neg 54.5%

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

   4. remove-double-neg 54.5%

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

   5. sub0-neg 54.5%

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

   6. distribute-neg-in 54.5%

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

   7. +-commutative 54.5%

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

   8. sub-neg 54.5%

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

   9. cos-neg 54.5%

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

   10. associate-*l* 54.5%

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

   11. distribute-rgt-neg-in 54.5%

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

   12. *-commutative 54.5%

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

3. Simplified 54.5%

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

4. Taylor expanded in im around 0 52.1%

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

   1. log1p-expm1-u 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*r* 99.3%

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

   5. metadata-eval 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 2: 89.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5600:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{3} + im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot \left(\cos re \cdot {im}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5600.0)
   (* 0.5 (* (cos re) (+ (* -0.3333333333333333 (pow im 3.0)) (* im -2.0))))
   (if (<= im 5.6e+102)
     (log1p (expm1 (- im)))
     (* 0.5 (* -0.3333333333333333 (* (cos re) (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5600.0) {
		tmp = 0.5 * (cos(re) * ((-0.3333333333333333 * pow(im, 3.0)) + (im * -2.0)));
	} else if (im <= 5.6e+102) {
		tmp = log1p(expm1(-im));
	} else {
		tmp = 0.5 * (-0.3333333333333333 * (cos(re) * pow(im, 3.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5600.0) {
		tmp = 0.5 * (Math.cos(re) * ((-0.3333333333333333 * Math.pow(im, 3.0)) + (im * -2.0)));
	} else if (im <= 5.6e+102) {
		tmp = Math.log1p(Math.expm1(-im));
	} else {
		tmp = 0.5 * (-0.3333333333333333 * (Math.cos(re) * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5600.0:
		tmp = 0.5 * (math.cos(re) * ((-0.3333333333333333 * math.pow(im, 3.0)) + (im * -2.0)))
	elif im <= 5.6e+102:
		tmp = math.log1p(math.expm1(-im))
	else:
		tmp = 0.5 * (-0.3333333333333333 * (math.cos(re) * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5600.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(-0.3333333333333333 * (im ^ 3.0)) + Float64(im * -2.0))));
	elseif (im <= 5.6e+102)
		tmp = log1p(expm1(Float64(-im)));
	else
		tmp = Float64(0.5 * Float64(-0.3333333333333333 * Float64(cos(re) * (im ^ 3.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5600.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6e+102], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(-0.3333333333333333 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5600

   1. Initial program 42.1%

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

      1. sub-neg 42.1%

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

      2. neg-sub0 42.1%

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

      3. remove-double-neg 42.1%

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

      4. remove-double-neg 42.1%

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

      5. sub0-neg 42.1%

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

      6. distribute-neg-in 42.1%

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

      7. +-commutative 42.1%

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

      8. sub-neg 42.1%

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

      9. cos-neg 42.1%

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

      10. associate-*l* 42.1%

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

      11. distribute-rgt-neg-in 42.1%

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

      12. *-commutative 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in im around 0 86.7%

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

   if 5600 < im < 5.60000000000000037e102

   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. Taylor expanded in im around 0 3.6%

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

   5. Taylor expanded in re around 0 2.5%

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

      1. *-commutative 2.5%

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

   7. Simplified 2.5%

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

      1. log1p-expm1-u 61.5%

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

      2. *-commutative 61.5%

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

      3. associate-*l* 61.5%

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

      4. metadata-eval 61.5%

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

   9. Applied egg-rr 61.5%

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

   if 5.60000000000000037e102 < 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. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5600:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{3} + im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot \left(\cos re \cdot {im}^{3}\right)\right)\\ \end{array} \]

Alternative 3: 72.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5600:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot \left(\cos re \cdot {im}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5600.0)
   (* im (- (cos re)))
   (if (<= im 5.6e+102)
     (log1p (expm1 (- im)))
     (* 0.5 (* -0.3333333333333333 (* (cos re) (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5600.0) {
		tmp = im * -cos(re);
	} else if (im <= 5.6e+102) {
		tmp = log1p(expm1(-im));
	} else {
		tmp = 0.5 * (-0.3333333333333333 * (cos(re) * pow(im, 3.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5600.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 5.6e+102) {
		tmp = Math.log1p(Math.expm1(-im));
	} else {
		tmp = 0.5 * (-0.3333333333333333 * (Math.cos(re) * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5600.0:
		tmp = im * -math.cos(re)
	elif im <= 5.6e+102:
		tmp = math.log1p(math.expm1(-im))
	else:
		tmp = 0.5 * (-0.3333333333333333 * (math.cos(re) * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5600.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 5.6e+102)
		tmp = log1p(expm1(Float64(-im)));
	else
		tmp = Float64(0.5 * Float64(-0.3333333333333333 * Float64(cos(re) * (im ^ 3.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5600.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 5.6e+102], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(-0.3333333333333333 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5600

   1. Initial program 42.1%

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

      1. sub-neg 42.1%

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

      2. neg-sub0 42.1%

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

      3. remove-double-neg 42.1%

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

      4. remove-double-neg 42.1%

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

      5. sub0-neg 42.1%

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

      6. distribute-neg-in 42.1%

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

      7. +-commutative 42.1%

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

      8. sub-neg 42.1%

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

      9. cos-neg 42.1%

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

      10. associate-*l* 42.1%

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

      11. distribute-rgt-neg-in 42.1%

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

      12. *-commutative 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in im around 0 64.8%

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

   5. Taylor expanded in im around 0 64.4%

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

      1. associate-*r* 64.4%

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

      2. *-commutative 64.4%

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

      3. mul-1-neg 64.4%

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

   7. Simplified 64.4%

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

   if 5600 < im < 5.60000000000000037e102

   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. Taylor expanded in im around 0 3.6%

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

   5. Taylor expanded in re around 0 2.5%

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

      1. *-commutative 2.5%

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

   7. Simplified 2.5%

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

      1. log1p-expm1-u 61.5%

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

      2. *-commutative 61.5%

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

      3. associate-*l* 61.5%

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

      4. metadata-eval 61.5%

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

   9. Applied egg-rr 61.5%

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

   if 5.60000000000000037e102 < 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. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5600:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot \left(\cos re \cdot {im}^{3}\right)\right)\\ \end{array} \]

Alternative 4: 68.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5600:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5600.0) (* im (- (cos re))) (log1p (expm1 (- im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5600.0) {
		tmp = im * -cos(re);
	} else {
		tmp = log1p(expm1(-im));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5600.0) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = Math.log1p(Math.expm1(-im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5600.0:
		tmp = im * -math.cos(re)
	else:
		tmp = math.log1p(math.expm1(-im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5600.0)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = log1p(expm1(Float64(-im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5600.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5600

   1. Initial program 42.1%

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

      1. sub-neg 42.1%

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

      2. neg-sub0 42.1%

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

      3. remove-double-neg 42.1%

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

      4. remove-double-neg 42.1%

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

      5. sub0-neg 42.1%

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

      6. distribute-neg-in 42.1%

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

      7. +-commutative 42.1%

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

      8. sub-neg 42.1%

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

      9. cos-neg 42.1%

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

      10. associate-*l* 42.1%

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

      11. distribute-rgt-neg-in 42.1%

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

      12. *-commutative 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in im around 0 64.8%

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

   5. Taylor expanded in im around 0 64.4%

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

      1. associate-*r* 64.4%

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

      2. *-commutative 64.4%

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

      3. mul-1-neg 64.4%

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

   7. Simplified 64.4%

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

   if 5600 < 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. Taylor expanded in im around 0 5.9%

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

   5. Taylor expanded in re around 0 5.1%

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

      1. *-commutative 5.1%

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

   7. Simplified 5.1%

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

      1. log1p-expm1-u 80.0%

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

      2. *-commutative 80.0%

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

      3. associate-*l* 80.0%

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

      4. metadata-eval 80.0%

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

   9. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5600:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \end{array} \]

Alternative 5: 63.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 600:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3} + im \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 600.0)
   (* im (- (cos re)))
   (if (<= im 1.6e+75)
     (* 0.5 (* im (+ -2.0 (pow re 2.0))))
     (* 0.5 (+ (* -0.3333333333333333 (pow im 3.0)) (* im -2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 600.0) {
		tmp = im * -cos(re);
	} else if (im <= 1.6e+75) {
		tmp = 0.5 * (im * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = 0.5 * ((-0.3333333333333333 * pow(im, 3.0)) + (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 (im <= 600.0d0) then
        tmp = im * -cos(re)
    else if (im <= 1.6d+75) then
        tmp = 0.5d0 * (im * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = 0.5d0 * (((-0.3333333333333333d0) * (im ** 3.0d0)) + (im * (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 600.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.6e+75) {
		tmp = 0.5 * (im * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * ((-0.3333333333333333 * Math.pow(im, 3.0)) + (im * -2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 600.0:
		tmp = im * -math.cos(re)
	elif im <= 1.6e+75:
		tmp = 0.5 * (im * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = 0.5 * ((-0.3333333333333333 * math.pow(im, 3.0)) + (im * -2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 600.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.6e+75)
		tmp = Float64(0.5 * Float64(im * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(Float64(-0.3333333333333333 * (im ^ 3.0)) + Float64(im * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 600.0)
		tmp = im * -cos(re);
	elseif (im <= 1.6e+75)
		tmp = 0.5 * (im * (-2.0 + (re ^ 2.0)));
	else
		tmp = 0.5 * ((-0.3333333333333333 * (im ^ 3.0)) + (im * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 600.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.6e+75], N[(0.5 * N[(im * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 600

   1. Initial program 41.8%

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

      1. sub-neg 41.8%

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

      2. neg-sub0 41.8%

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

      3. remove-double-neg 41.8%

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

      4. remove-double-neg 41.8%

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

      5. sub0-neg 41.8%

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

      6. distribute-neg-in 41.8%

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

      7. +-commutative 41.8%

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

      8. sub-neg 41.8%

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

      9. cos-neg 41.8%

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

      10. associate-*l* 41.8%

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

      11. distribute-rgt-neg-in 41.8%

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

      12. *-commutative 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in im around 0 65.1%

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

   5. Taylor expanded in im around 0 64.7%

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

      1. associate-*r* 64.7%

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

      2. *-commutative 64.7%

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

      3. mul-1-neg 64.7%

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

   7. Simplified 64.7%

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

   if 600 < im < 1.59999999999999992e75

   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. Taylor expanded in im around 0 3.4%

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

   5. Taylor expanded in re around 0 42.5%

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

      1. +-commutative 42.5%

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

      2. *-commutative 42.5%

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

      3. distribute-lft-out 42.5%

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

   7. Simplified 42.5%

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

   if 1.59999999999999992e75 < 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. Taylor expanded in im around 0 92.0%

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

   5. Taylor expanded in re around 0 78.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 600:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3} + im \cdot -2\right)\\ \end{array} \]

Alternative 6: 55.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 380:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 380.0) (* im (- (cos re))) (* 0.5 (* im (+ -2.0 (pow re 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 380.0) {
		tmp = im * -cos(re);
	} else {
		tmp = 0.5 * (im * (-2.0 + pow(re, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 380.0d0) then
        tmp = im * -cos(re)
    else
        tmp = 0.5d0 * (im * ((-2.0d0) + (re ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 380.0) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = 0.5 * (im * (-2.0 + Math.pow(re, 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 380.0:
		tmp = im * -math.cos(re)
	else:
		tmp = 0.5 * (im * (-2.0 + math.pow(re, 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 380.0)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(0.5 * Float64(im * Float64(-2.0 + (re ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 380.0)
		tmp = im * -cos(re);
	else
		tmp = 0.5 * (im * (-2.0 + (re ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 380.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(0.5 * N[(im * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 380

   1. Initial program 41.8%

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

      1. sub-neg 41.8%

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

      2. neg-sub0 41.8%

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

      3. remove-double-neg 41.8%

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

      4. remove-double-neg 41.8%

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

      5. sub0-neg 41.8%

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

      6. distribute-neg-in 41.8%

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

      7. +-commutative 41.8%

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

      8. sub-neg 41.8%

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

      9. cos-neg 41.8%

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

      10. associate-*l* 41.8%

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

      11. distribute-rgt-neg-in 41.8%

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

      12. *-commutative 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in im around 0 65.1%

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

   5. Taylor expanded in im around 0 64.7%

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

      1. associate-*r* 64.7%

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

      2. *-commutative 64.7%

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

      3. mul-1-neg 64.7%

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

   7. Simplified 64.7%

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

   if 380 < 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. Taylor expanded in im around 0 5.9%

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

   5. Taylor expanded in re around 0 17.9%

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

      1. +-commutative 17.9%

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

      2. *-commutative 17.9%

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

      3. distribute-lft-out 17.9%

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

   7. Simplified 17.9%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 380:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \end{array} \]

Alternative 7: 55.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 760:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 760.0) (* im (- (cos re))) (* 0.5 (* im (pow re 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 760.0) {
		tmp = im * -cos(re);
	} else {
		tmp = 0.5 * (im * pow(re, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 760.0d0) then
        tmp = im * -cos(re)
    else
        tmp = 0.5d0 * (im * (re ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 760.0) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = 0.5 * (im * Math.pow(re, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 760.0:
		tmp = im * -math.cos(re)
	else:
		tmp = 0.5 * (im * math.pow(re, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 760.0)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 760.0)
		tmp = im * -cos(re);
	else
		tmp = 0.5 * (im * (re ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 760.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 760

   1. Initial program 41.8%

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

      1. sub-neg 41.8%

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

      2. neg-sub0 41.8%

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

      3. remove-double-neg 41.8%

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

      4. remove-double-neg 41.8%

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

      5. sub0-neg 41.8%

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

      6. distribute-neg-in 41.8%

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

      7. +-commutative 41.8%

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

      8. sub-neg 41.8%

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

      9. cos-neg 41.8%

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

      10. associate-*l* 41.8%

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

      11. distribute-rgt-neg-in 41.8%

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

      12. *-commutative 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in im around 0 65.1%

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

   5. Taylor expanded in im around 0 64.7%

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

      1. associate-*r* 64.7%

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

      2. *-commutative 64.7%

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

      3. mul-1-neg 64.7%

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

   7. Simplified 64.7%

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

   if 760 < 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. Taylor expanded in im around 0 5.9%

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

   5. Taylor expanded in re around 0 17.9%

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

      1. +-commutative 17.9%

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

      2. *-commutative 17.9%

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

      3. distribute-lft-out 17.9%

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

   7. Simplified 17.9%

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

   8. Taylor expanded in re around inf 15.3%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 760:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right)\\ \end{array} \]

Alternative 8: 51.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

   1. sub-neg 54.5%

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

   2. neg-sub0 54.5%

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

   3. remove-double-neg 54.5%

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

   4. remove-double-neg 54.5%

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

   5. sub0-neg 54.5%

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

   6. distribute-neg-in 54.5%

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

   7. +-commutative 54.5%

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

   8. sub-neg 54.5%

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

   9. cos-neg 54.5%

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

   10. associate-*l* 54.5%

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

   11. distribute-rgt-neg-in 54.5%

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

   12. *-commutative 54.5%

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

3. Simplified 54.5%

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

4. Taylor expanded in im around 0 52.1%

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

5. Taylor expanded in im around 0 51.8%

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

   1. associate-*r* 51.8%

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

   2. *-commutative 51.8%

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

   3. mul-1-neg 51.8%

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

7. Simplified 51.8%

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

8. Final simplification 51.8%

   \[\leadsto im \cdot \left(-\cos re\right) \]

Alternative 9: 30.2% 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 54.5%

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

   1. sub-neg 54.5%

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

   2. neg-sub0 54.5%

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

   3. remove-double-neg 54.5%

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

   4. remove-double-neg 54.5%

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

   5. sub0-neg 54.5%

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

   6. distribute-neg-in 54.5%

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

   7. +-commutative 54.5%

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

   8. sub-neg 54.5%

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

   9. cos-neg 54.5%

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

   10. associate-*l* 54.5%

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

   11. distribute-rgt-neg-in 54.5%

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

   12. *-commutative 54.5%

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

3. Simplified 54.5%

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

4. Taylor expanded in im around 0 52.1%

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

5. Taylor expanded in re around 0 29.1%

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

   1. *-commutative 29.1%

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

7. Simplified 29.1%

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

8. Final simplification 29.1%

   \[\leadsto 0.5 \cdot \left(im \cdot -2\right) \]

Alternative 10: 30.1% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

   1. sub-neg 54.5%

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

   2. neg-sub0 54.5%

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

   3. remove-double-neg 54.5%

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

   4. remove-double-neg 54.5%

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

   5. sub0-neg 54.5%

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

   6. distribute-neg-in 54.5%

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

   7. +-commutative 54.5%

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

   8. sub-neg 54.5%

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

   9. cos-neg 54.5%

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

   10. associate-*l* 54.5%

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

   11. distribute-rgt-neg-in 54.5%

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

   12. *-commutative 54.5%

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

3. Simplified 54.5%

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

4. Taylor expanded in im around 0 52.1%

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

5. Taylor expanded in re around 0 28.8%

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

   1. mul-1-neg 28.8%

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

7. Simplified 28.8%

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

8. Final simplification 28.8%

   \[\leadsto -im \]

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 2023319 
(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))))