math.sin on complex, imaginary part

Percentage Accurate: 54.5% → 99.0%

Time: 11.5s

Alternatives: 11

Speedup: 2.8×

• 49.6% 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: 54.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.3%

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

   1. cos-neg 51.3%

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

   2. sub-neg 51.3%

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

   3. neg-sub0 51.3%

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

   4. remove-double-neg 51.3%

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

   5. remove-double-neg 51.3%

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

   6. sub0-neg 51.3%

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

   7. distribute-neg-in 51.3%

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

   8. +-commutative 51.3%

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

   9. sub-neg 51.3%

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

   10. associate-*l* 51.3%

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

   11. distribute-rgt-neg-in 51.3%

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

   12. *-commutative 51.3%

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

3. Simplified 51.3%

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

5. Taylor expanded in im around 0 55.3%

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

   1. log1p-expm1-u 99.2%

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

   2. associate-*r* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*r* 99.2%

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

   5. metadata-eval 99.2%

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

7. Applied egg-rr 99.2%

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

8. Taylor expanded in re around inf 50.6%

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

   1. expm1-define 99.2%

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

   2. mul-1-neg 99.2%

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

   3. distribute-rgt-neg-out 99.2%

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

10. Simplified 99.2%

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

11. Final simplification 99.2%

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

Alternative 2: 73.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 500:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 500.0)
   (* (cos re) (- im))
   (if (<= im 5.6e+102)
     (log1p (expm1 (- im)))
     (* 0.5 (* (cos re) (* -0.3333333333333333 (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 500.0) {
		tmp = cos(re) * -im;
	} else if (im <= 5.6e+102) {
		tmp = log1p(expm1(-im));
	} else {
		tmp = 0.5 * (cos(re) * (-0.3333333333333333 * pow(im, 3.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 500.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 5.6e+102) {
		tmp = Math.log1p(Math.expm1(-im));
	} else {
		tmp = 0.5 * (Math.cos(re) * (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 500.0:
		tmp = math.cos(re) * -im
	elif im <= 5.6e+102:
		tmp = math.log1p(math.expm1(-im))
	else:
		tmp = 0.5 * (math.cos(re) * (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 500.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 5.6e+102)
		tmp = log1p(expm1(Float64(-im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 500.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 5.6e+102], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 500

   1. Initial program 36.7%

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

      1. cos-neg 36.7%

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

      2. sub-neg 36.7%

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

      3. neg-sub0 36.7%

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

      4. remove-double-neg 36.7%

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

      5. remove-double-neg 36.7%

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

      6. sub0-neg 36.7%

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

      7. distribute-neg-in 36.7%

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

      8. +-commutative 36.7%

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

      9. sub-neg 36.7%

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

      10. associate-*l* 36.7%

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

      11. distribute-rgt-neg-in 36.7%

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

      12. *-commutative 36.7%

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

   3. Simplified 36.7%

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

   5. Taylor expanded in im around 0 69.3%

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

   6. Taylor expanded in im around 0 69.3%

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

      1. associate-*r* 69.3%

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

      2. *-commutative 69.3%

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

      3. mul-1-neg 69.3%

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

   8. Simplified 69.3%

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

   if 500 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. 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%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 81.0%

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

      1. expm1-define 81.0%

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

      2. mul-1-neg 81.0%

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

   10. Simplified 81.0%

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

   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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. 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%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 500:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.3333333333333333 \cdot {im}^{3}\\ \mathbf{if}\;im \leq 480:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.3333333333333333 (pow im 3.0))))
   (if (<= im 480.0)
     (* 0.5 (* (cos re) (+ (* im -2.0) t_0)))
     (if (<= im 5.6e+102) (log1p (expm1 (- im))) (* 0.5 (* (cos re) t_0))))))

C

double code(double re, double im) {
	double t_0 = -0.3333333333333333 * pow(im, 3.0);
	double tmp;
	if (im <= 480.0) {
		tmp = 0.5 * (cos(re) * ((im * -2.0) + t_0));
	} else if (im <= 5.6e+102) {
		tmp = log1p(expm1(-im));
	} else {
		tmp = 0.5 * (cos(re) * t_0);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = -0.3333333333333333 * Math.pow(im, 3.0);
	double tmp;
	if (im <= 480.0) {
		tmp = 0.5 * (Math.cos(re) * ((im * -2.0) + t_0));
	} else if (im <= 5.6e+102) {
		tmp = Math.log1p(Math.expm1(-im));
	} else {
		tmp = 0.5 * (Math.cos(re) * t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.3333333333333333 * math.pow(im, 3.0)
	tmp = 0
	if im <= 480.0:
		tmp = 0.5 * (math.cos(re) * ((im * -2.0) + t_0))
	elif im <= 5.6e+102:
		tmp = math.log1p(math.expm1(-im))
	else:
		tmp = 0.5 * (math.cos(re) * t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.3333333333333333 * (im ^ 3.0))
	tmp = 0.0
	if (im <= 480.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(im * -2.0) + t_0)));
	elseif (im <= 5.6e+102)
		tmp = log1p(expm1(Float64(-im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * t_0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 480.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(im * -2.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6e+102], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 480

   1. Initial program 36.7%

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

      1. cos-neg 36.7%

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

      2. sub-neg 36.7%

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

      3. neg-sub0 36.7%

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

      4. remove-double-neg 36.7%

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

      5. remove-double-neg 36.7%

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

      6. sub0-neg 36.7%

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

      7. distribute-neg-in 36.7%

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

      8. +-commutative 36.7%

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

      9. sub-neg 36.7%

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

      10. associate-*l* 36.7%

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

      11. distribute-rgt-neg-in 36.7%

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

      12. *-commutative 36.7%

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

   3. Simplified 36.7%

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

   5. Taylor expanded in im around 0 90.9%

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

   if 480 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. 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%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 81.0%

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

      1. expm1-define 81.0%

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

      2. mul-1-neg 81.0%

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

   10. Simplified 81.0%

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

   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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. 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%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 480:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3500:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+103}:\\ \;\;\;\;\log \left(e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3500.0)
   (* (cos re) (- im))
   (if (<= im 1.2e+103)
     (log (exp im))
     (* 0.5 (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3500.0) {
		tmp = cos(re) * -im;
	} else if (im <= 1.2e+103) {
		tmp = log(exp(im));
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3500.0d0) then
        tmp = cos(re) * -im
    else if (im <= 1.2d+103) then
        tmp = log(exp(im))
    else
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((-0.3333333333333333d0) * (im ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3500.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.2e+103) {
		tmp = Math.log(Math.exp(im));
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3500.0:
		tmp = math.cos(re) * -im
	elif im <= 1.2e+103:
		tmp = math.log(math.exp(im))
	else:
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3500.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.2e+103)
		tmp = log(exp(im));
	else
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3500.0)
		tmp = cos(re) * -im;
	elseif (im <= 1.2e+103)
		tmp = log(exp(im));
	else
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * (im ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3500.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.2e+103], N[Log[N[Exp[im], $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3500

   1. Initial program 37.4%

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

      1. cos-neg 37.4%

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

      2. sub-neg 37.4%

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

      3. neg-sub0 37.4%

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

      4. remove-double-neg 37.4%

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

      5. remove-double-neg 37.4%

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

      6. sub0-neg 37.4%

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

      7. distribute-neg-in 37.4%

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

      8. +-commutative 37.4%

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

      9. sub-neg 37.4%

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

      10. associate-*l* 37.4%

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

      11. distribute-rgt-neg-in 37.4%

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

      12. *-commutative 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in im around 0 68.7%

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

   6. Taylor expanded in im around 0 68.7%

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

      1. associate-*r* 68.7%

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

      2. *-commutative 68.7%

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

      3. mul-1-neg 68.7%

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

   8. Simplified 68.7%

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

   if 3500 < im < 1.1999999999999999e103

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. 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%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 75.0%

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

      1. expm1-define 75.0%

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

      2. mul-1-neg 75.0%

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

   10. Simplified 75.0%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-im\right)}\right) \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

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

       2. log1p-expm1-u 0.0%

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

       3. sqrt-unprod 1.3%

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

       4. sqr-neg 1.3%

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

       5. sqrt-prod 1.3%

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

       6. add-sqr-sqrt 1.3%

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

       7. add-log-exp 25.0%

          \[\leadsto \color{blue}{\log \left(e^{im}\right)} \]

   12. Applied egg-rr 25.0%

       \[\leadsto \color{blue}{\log \left(e^{im}\right)} \]

   if 1.1999999999999999e103 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. 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%

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

   6. Taylor expanded in re around 0 78.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3500:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+103}:\\ \;\;\;\;\log \left(e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 490:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 490.0) (* (cos re) (- im)) (log1p (expm1 (- im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 490.0) {
		tmp = cos(re) * -im;
	} else {
		tmp = log1p(expm1(-im));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 490.0) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = Math.log1p(Math.expm1(-im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 490.0:
		tmp = math.cos(re) * -im
	else:
		tmp = math.log1p(math.expm1(-im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 490.0)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = log1p(expm1(Float64(-im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 490.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 490

   1. Initial program 36.7%

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

      1. cos-neg 36.7%

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

      2. sub-neg 36.7%

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

      3. neg-sub0 36.7%

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

      4. remove-double-neg 36.7%

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

      5. remove-double-neg 36.7%

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

      6. sub0-neg 36.7%

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

      7. distribute-neg-in 36.7%

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

      8. +-commutative 36.7%

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

      9. sub-neg 36.7%

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

      10. associate-*l* 36.7%

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

      11. distribute-rgt-neg-in 36.7%

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

      12. *-commutative 36.7%

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

   3. Simplified 36.7%

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

   5. Taylor expanded in im around 0 69.3%

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

   6. Taylor expanded in im around 0 69.3%

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

      1. associate-*r* 69.3%

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

      2. *-commutative 69.3%

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

      3. mul-1-neg 69.3%

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

   8. Simplified 69.3%

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

   if 490 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. 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 8.4%

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

      1. log1p-expm1-u 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 78.0%

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

      1. expm1-define 78.0%

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

      2. mul-1-neg 78.0%

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

   10. Simplified 78.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 490:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2350:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+91}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right) - im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2350.0)
   (* (cos re) (- im))
   (if (<= im 1.55e+91)
     (- (* 0.5 (* im (pow re 2.0))) im)
     (* 0.5 (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2350.0) {
		tmp = cos(re) * -im;
	} else if (im <= 1.55e+91) {
		tmp = (0.5 * (im * pow(re, 2.0))) - im;
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2350.0d0) then
        tmp = cos(re) * -im
    else if (im <= 1.55d+91) then
        tmp = (0.5d0 * (im * (re ** 2.0d0))) - im
    else
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((-0.3333333333333333d0) * (im ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2350.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.55e+91) {
		tmp = (0.5 * (im * Math.pow(re, 2.0))) - im;
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2350.0:
		tmp = math.cos(re) * -im
	elif im <= 1.55e+91:
		tmp = (0.5 * (im * math.pow(re, 2.0))) - im
	else:
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2350.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.55e+91)
		tmp = Float64(Float64(0.5 * Float64(im * (re ^ 2.0))) - im);
	else
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2350.0)
		tmp = cos(re) * -im;
	elseif (im <= 1.55e+91)
		tmp = (0.5 * (im * (re ^ 2.0))) - im;
	else
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * (im ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2350.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.55e+91], N[(N[(0.5 * N[(im * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2350

   1. Initial program 37.4%

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

      1. cos-neg 37.4%

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

      2. sub-neg 37.4%

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

      3. neg-sub0 37.4%

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

      4. remove-double-neg 37.4%

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

      5. remove-double-neg 37.4%

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

      6. sub0-neg 37.4%

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

      7. distribute-neg-in 37.4%

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

      8. +-commutative 37.4%

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

      9. sub-neg 37.4%

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

      10. associate-*l* 37.4%

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

      11. distribute-rgt-neg-in 37.4%

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

      12. *-commutative 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in im around 0 68.7%

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

   6. Taylor expanded in im around 0 68.7%

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

      1. associate-*r* 68.7%

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

      2. *-commutative 68.7%

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

      3. mul-1-neg 68.7%

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

   8. Simplified 68.7%

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

   if 2350 < im < 1.54999999999999999e91

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. 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. Taylor expanded in re around 0 18.0%

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

      1. +-commutative 18.0%

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

      2. mul-1-neg 18.0%

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

      3. unsub-neg 18.0%

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

   8. Simplified 18.0%

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

   if 1.54999999999999999e91 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. 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%

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

   6. Taylor expanded in re around 0 76.3%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2350:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+91}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right) - im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2350:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+137}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right) - im\\ \mathbf{else}:\\ \;\;\;\;\frac{-{im}^{2}}{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2350.0)
   (* (cos re) (- im))
   (if (<= im 1.15e+137)
     (- (* 0.5 (* im (pow re 2.0))) im)
     (/ (- (pow im 2.0)) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2350.0) {
		tmp = cos(re) * -im;
	} else if (im <= 1.15e+137) {
		tmp = (0.5 * (im * pow(re, 2.0))) - im;
	} else {
		tmp = -pow(im, 2.0) / im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2350.0d0) then
        tmp = cos(re) * -im
    else if (im <= 1.15d+137) then
        tmp = (0.5d0 * (im * (re ** 2.0d0))) - im
    else
        tmp = -(im ** 2.0d0) / im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2350.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.15e+137) {
		tmp = (0.5 * (im * Math.pow(re, 2.0))) - im;
	} else {
		tmp = -Math.pow(im, 2.0) / im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2350.0:
		tmp = math.cos(re) * -im
	elif im <= 1.15e+137:
		tmp = (0.5 * (im * math.pow(re, 2.0))) - im
	else:
		tmp = -math.pow(im, 2.0) / im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2350.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.15e+137)
		tmp = Float64(Float64(0.5 * Float64(im * (re ^ 2.0))) - im);
	else
		tmp = Float64(Float64(-(im ^ 2.0)) / im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2350.0)
		tmp = cos(re) * -im;
	elseif (im <= 1.15e+137)
		tmp = (0.5 * (im * (re ^ 2.0))) - im;
	else
		tmp = -(im ^ 2.0) / im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2350.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.15e+137], N[(N[(0.5 * N[(im * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[((-N[Power[im, 2.0], $MachinePrecision]) / im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2350

   1. Initial program 37.4%

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

      1. cos-neg 37.4%

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

      2. sub-neg 37.4%

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

      3. neg-sub0 37.4%

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

      4. remove-double-neg 37.4%

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

      5. remove-double-neg 37.4%

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

      6. sub0-neg 37.4%

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

      7. distribute-neg-in 37.4%

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

      8. +-commutative 37.4%

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

      9. sub-neg 37.4%

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

      10. associate-*l* 37.4%

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

      11. distribute-rgt-neg-in 37.4%

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

      12. *-commutative 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in im around 0 68.7%

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

   6. Taylor expanded in im around 0 68.7%

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

      1. associate-*r* 68.7%

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

      2. *-commutative 68.7%

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

      3. mul-1-neg 68.7%

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

   8. Simplified 68.7%

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

   if 2350 < im < 1.15e137

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. 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 3.7%

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

   6. Taylor expanded in re around 0 14.1%

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

      1. +-commutative 14.1%

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

      2. mul-1-neg 14.1%

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

      3. unsub-neg 14.1%

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

   8. Simplified 14.1%

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

   if 1.15e137 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. 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 12.6%

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

      1. log1p-expm1-u 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 77.4%

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

      1. expm1-define 77.4%

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

      2. mul-1-neg 77.4%

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

   10. Simplified 77.4%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-im\right)}\right) \]
   11. Step-by-step derivation

       1. log1p-expm1-u 5.6%

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

       2. neg-sub0 5.6%

          \[\leadsto \color{blue}{0 - im} \]

       3. flip-- 68.2%

          \[\leadsto \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

       4. metadata-eval 68.2%

          \[\leadsto \frac{\color{blue}{0} - im \cdot im}{0 + im} \]

       5. unpow2 68.2%

          \[\leadsto \frac{0 - \color{blue}{{im}^{2}}}{0 + im} \]

       6. add-sqr-sqrt 68.2%

          \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{im} \cdot \sqrt{im}}} \]

       7. sqrt-prod 0.4%

          \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{im \cdot im}}} \]

       8. sqr-neg 0.4%

          \[\leadsto \frac{0 - {im}^{2}}{0 + \sqrt{\color{blue}{\left(-im\right) \cdot \left(-im\right)}}} \]

       9. sqrt-unprod 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} \]

       10. add-sqr-sqrt 22.6%

           \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\left(-im\right)}} \]

       11. sub-neg 22.6%

           \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{0 - im}} \]

       12. neg-sub0 22.6%

           \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{-im}} \]

       13. add-sqr-sqrt 0.0%

           \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} \]

       14. sqrt-unprod 0.4%

           \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}} \]

       15. sqr-neg 0.4%

           \[\leadsto \frac{0 - {im}^{2}}{\sqrt{\color{blue}{im \cdot im}}} \]

       16. sqrt-prod 68.2%

           \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{im} \cdot \sqrt{im}}} \]

       17. add-sqr-sqrt 68.2%

           \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{im}} \]

   12. Applied egg-rr 68.2%

       \[\leadsto \color{blue}{\frac{0 - {im}^{2}}{im}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2350:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+137}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right) - im\\ \mathbf{else}:\\ \;\;\;\;\frac{-{im}^{2}}{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -1 \cdot 10^{-310}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -1e-310) im (* 0.5 (* im -2.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -1e-310) {
		tmp = im;
	} else {
		tmp = 0.5 * (im * -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= (-1d-310)) then
        tmp = im
    else
        tmp = 0.5d0 * (im * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= -1e-310) {
		tmp = im;
	} else {
		tmp = 0.5 * (im * -2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= -1e-310:
		tmp = im
	else:
		tmp = 0.5 * (im * -2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -1e-310)
		tmp = im;
	else
		tmp = Float64(0.5 * Float64(im * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= -1e-310)
		tmp = im;
	else
		tmp = 0.5 * (im * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -1e-310], im, N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -9.999999999999969e-311

   1. Initial program 47.5%

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

      1. cos-neg 47.5%

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

      2. sub-neg 47.5%

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

      3. neg-sub0 47.5%

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

      4. remove-double-neg 47.5%

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

      5. remove-double-neg 47.5%

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

      6. sub0-neg 47.5%

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

      7. distribute-neg-in 47.5%

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

      8. +-commutative 47.5%

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

      9. sub-neg 47.5%

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

      10. associate-*l* 47.5%

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

      11. distribute-rgt-neg-in 47.5%

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

      12. *-commutative 47.5%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in im around 0 59.3%

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

      1. log1p-expm1-u 98.6%

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

      2. associate-*r* 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-*r* 98.6%

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

      5. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in re around 0 2.9%

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

      1. expm1-define 2.1%

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

      2. mul-1-neg 2.1%

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

   10. Simplified 2.1%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-im\right)}\right) \]
   11. Step-by-step derivation

       1. neg-sub0 2.1%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{0 - im}\right)\right) \]

       2. log1p-expm1-u 2.2%

          \[\leadsto \color{blue}{0 - im} \]

       3. sub-neg 2.2%

          \[\leadsto \color{blue}{0 + \left(-im\right)} \]

       4. add-sqr-sqrt 1.1%

          \[\leadsto 0 + \color{blue}{\sqrt{-im} \cdot \sqrt{-im}} \]

       5. sqrt-unprod 16.7%

          \[\leadsto 0 + \color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} \]

       6. sqr-neg 16.7%

          \[\leadsto 0 + \sqrt{\color{blue}{im \cdot im}} \]

       7. sqrt-prod 8.5%

          \[\leadsto 0 + \color{blue}{\sqrt{im} \cdot \sqrt{im}} \]

       8. add-sqr-sqrt 14.9%

          \[\leadsto 0 + \color{blue}{im} \]

   12. Applied egg-rr 14.9%

       \[\leadsto \color{blue}{0 + im} \]

   if -9.999999999999969e-311 < (cos.f64 re)

   1. Initial program 52.9%

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

      1. cos-neg 52.9%

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

      2. sub-neg 52.9%

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

      3. neg-sub0 52.9%

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

      4. remove-double-neg 52.9%

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

      5. remove-double-neg 52.9%

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

      6. sub0-neg 52.9%

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

      7. distribute-neg-in 52.9%

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

      8. +-commutative 52.9%

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

      9. sub-neg 52.9%

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

      10. associate-*l* 52.9%

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

      11. distribute-rgt-neg-in 52.9%

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

      12. *-commutative 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in im around 0 53.6%

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

   6. Taylor expanded in re around 0 40.8%

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

      1. *-commutative 40.8%

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

   8. Simplified 40.8%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -1 \cdot 10^{-310}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.15 \cdot 10^{+137}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-{im}^{2}}{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.15e+137) (* (cos re) (- im)) (/ (- (pow im 2.0)) im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.15e+137) {
		tmp = cos(re) * -im;
	} else {
		tmp = -pow(im, 2.0) / im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.15d+137) then
        tmp = cos(re) * -im
    else
        tmp = -(im ** 2.0d0) / im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.15e+137) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = -Math.pow(im, 2.0) / im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.15e+137:
		tmp = math.cos(re) * -im
	else:
		tmp = -math.pow(im, 2.0) / im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.15e+137)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = Float64(Float64(-(im ^ 2.0)) / im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.15e+137)
		tmp = cos(re) * -im;
	else
		tmp = -(im ^ 2.0) / im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.15e+137], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], N[((-N[Power[im, 2.0], $MachinePrecision]) / im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.15e137

   1. Initial program 44.6%

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

      1. cos-neg 44.6%

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

      2. sub-neg 44.6%

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

      3. neg-sub0 44.6%

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

      4. remove-double-neg 44.6%

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

      5. remove-double-neg 44.6%

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

      6. sub0-neg 44.6%

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

      7. distribute-neg-in 44.6%

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

      8. +-commutative 44.6%

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

      9. sub-neg 44.6%

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

      10. associate-*l* 44.6%

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

      11. distribute-rgt-neg-in 44.6%

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

      12. *-commutative 44.6%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in im around 0 61.2%

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

   6. Taylor expanded in im around 0 61.2%

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

      1. associate-*r* 61.2%

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

      2. *-commutative 61.2%

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

      3. mul-1-neg 61.2%

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

   8. Simplified 61.2%

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

   if 1.15e137 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. 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 12.6%

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

      1. log1p-expm1-u 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 77.4%

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

      1. expm1-define 77.4%

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

      2. mul-1-neg 77.4%

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

   10. Simplified 77.4%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-im\right)}\right) \]
   11. Step-by-step derivation

       1. log1p-expm1-u 5.6%

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

       2. neg-sub0 5.6%

          \[\leadsto \color{blue}{0 - im} \]

       3. flip-- 68.2%

          \[\leadsto \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \]

       4. metadata-eval 68.2%

          \[\leadsto \frac{\color{blue}{0} - im \cdot im}{0 + im} \]

       5. unpow2 68.2%

          \[\leadsto \frac{0 - \color{blue}{{im}^{2}}}{0 + im} \]

       6. add-sqr-sqrt 68.2%

          \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{im} \cdot \sqrt{im}}} \]

       7. sqrt-prod 0.4%

          \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{im \cdot im}}} \]

       8. sqr-neg 0.4%

          \[\leadsto \frac{0 - {im}^{2}}{0 + \sqrt{\color{blue}{\left(-im\right) \cdot \left(-im\right)}}} \]

       9. sqrt-unprod 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} \]

       10. add-sqr-sqrt 22.6%

           \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\left(-im\right)}} \]

       11. sub-neg 22.6%

           \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{0 - im}} \]

       12. neg-sub0 22.6%

           \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{-im}} \]

       13. add-sqr-sqrt 0.0%

           \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} \]

       14. sqrt-unprod 0.4%

           \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}} \]

       15. sqr-neg 0.4%

           \[\leadsto \frac{0 - {im}^{2}}{\sqrt{\color{blue}{im \cdot im}}} \]

       16. sqrt-prod 68.2%

           \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{im} \cdot \sqrt{im}}} \]

       17. add-sqr-sqrt 68.2%

           \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{im}} \]

   12. Applied egg-rr 68.2%

       \[\leadsto \color{blue}{\frac{0 - {im}^{2}}{im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.15 \cdot 10^{+137}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-{im}^{2}}{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.3%

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

   1. cos-neg 51.3%

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

   2. sub-neg 51.3%

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

   3. neg-sub0 51.3%

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

   4. remove-double-neg 51.3%

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

   5. remove-double-neg 51.3%

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

   6. sub0-neg 51.3%

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

   7. distribute-neg-in 51.3%

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

   8. +-commutative 51.3%

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

   9. sub-neg 51.3%

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

   10. associate-*l* 51.3%

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

   11. distribute-rgt-neg-in 51.3%

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

   12. *-commutative 51.3%

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

3. Simplified 51.3%

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

5. Taylor expanded in im around 0 55.3%

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

6. Taylor expanded in im around 0 54.7%

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

   1. associate-*r* 54.7%

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

   2. *-commutative 54.7%

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

   3. mul-1-neg 54.7%

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

8. Simplified 54.7%

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

9. Final simplification 54.7%

   \[\leadsto \cos re \cdot \left(-im\right) \]
10. Add Preprocessing

Alternative 11: 30.0% 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 51.3%

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

   1. cos-neg 51.3%

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

   2. sub-neg 51.3%

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

   3. neg-sub0 51.3%

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

   4. remove-double-neg 51.3%

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

   5. remove-double-neg 51.3%

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

   6. sub0-neg 51.3%

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

   7. distribute-neg-in 51.3%

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

   8. +-commutative 51.3%

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

   9. sub-neg 51.3%

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

   10. associate-*l* 51.3%

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

   11. distribute-rgt-neg-in 51.3%

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

   12. *-commutative 51.3%

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

3. Simplified 51.3%

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

5. Taylor expanded in im around 0 55.3%

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

6. Taylor expanded in re around 0 29.2%

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

   1. mul-1-neg 29.2%

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

8. Simplified 29.2%

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

9. Final simplification 29.2%

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

Developer target: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024030 
(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))))