math.sin on complex, imaginary part

Percentage Accurate: 53.9% → 99.2%

Time: 10.1s

Alternatives: 11

Speedup: 2.8×

• 49.4% 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: 53.9% 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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. cos-neg 54.2%

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

   2. sub-neg 54.2%

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

   3. neg-sub0 54.2%

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

   4. remove-double-neg 54.2%

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

   5. remove-double-neg 54.2%

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

   6. sub0-neg 54.2%

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

   7. distribute-neg-in 54.2%

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

   8. +-commutative 54.2%

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

   9. sub-neg 54.2%

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

   10. associate-*l* 54.2%

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

   11. sub-neg 54.2%

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

   12. +-commutative 54.2%

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

   13. distribute-neg-in 54.2%

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

3. Simplified 54.2%

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

5. Taylor expanded in im around 0 52.6%

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

   1. log1p-expm1-u 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 53.6%

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

   1. expm1-def 99.2%

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

   2. associate-*r* 99.2%

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

   3. mul-1-neg 99.2%

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

10. Simplified 99.2%

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

11. Final simplification 99.2%

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

Alternative 2: 74.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.7:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.7)
   (* (- im) (cos re))
   (if (<= im 1.1e+44)
     (log1p (expm1 (- im)))
     (* 0.5 (* (cos re) (* -0.0003968253968253968 (pow im 7.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.7) {
		tmp = -im * cos(re);
	} else if (im <= 1.1e+44) {
		tmp = log1p(expm1(-im));
	} else {
		tmp = 0.5 * (cos(re) * (-0.0003968253968253968 * pow(im, 7.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.7) {
		tmp = -im * Math.cos(re);
	} else if (im <= 1.1e+44) {
		tmp = Math.log1p(Math.expm1(-im));
	} else {
		tmp = 0.5 * (Math.cos(re) * (-0.0003968253968253968 * Math.pow(im, 7.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.7:
		tmp = -im * math.cos(re)
	elif im <= 1.1e+44:
		tmp = math.log1p(math.expm1(-im))
	else:
		tmp = 0.5 * (math.cos(re) * (-0.0003968253968253968 * math.pow(im, 7.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.7)
		tmp = Float64(Float64(-im) * cos(re));
	elseif (im <= 1.1e+44)
		tmp = log1p(expm1(Float64(-im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-0.0003968253968253968 * (im ^ 7.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.7], N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.69999999999999996

   1. Initial program 36.0%

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

      1. cos-neg 36.0%

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

      2. sub-neg 36.0%

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

      3. neg-sub0 36.0%

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

      4. remove-double-neg 36.0%

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

      5. remove-double-neg 36.0%

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

      6. sub0-neg 36.0%

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

      7. distribute-neg-in 36.0%

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

      8. +-commutative 36.0%

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

      9. sub-neg 36.0%

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

      10. associate-*l* 36.0%

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

      11. sub-neg 36.0%

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

      12. +-commutative 36.0%

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

      13. distribute-neg-in 36.0%

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

   3. Simplified 36.0%

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

   5. Taylor expanded in im around 0 70.9%

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

   6. Taylor expanded in im around 0 70.9%

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

      1. associate-*r* 70.9%

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

      2. *-commutative 70.9%

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

      3. mul-1-neg 70.9%

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

   8. Simplified 70.9%

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

   if 1.69999999999999996 < im < 1.09999999999999998e44

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

      1. log1p-expm1-u 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 80.0%

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

      1. expm1-def 80.0%

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

      2. mul-1-neg 80.0%

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

   10. Simplified 80.0%

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

   if 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.0003968253968253968 \cdot {im}^{7}\right) \cdot \cos re\right)} \]

      2. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)} \]

   8. Simplified 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.7:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 480:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 480.0)
   (* 0.5 (* (cos re) (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0)))))
   (if (<= im 1.1e+44)
     (log1p (expm1 (- im)))
     (* 0.5 (* (cos re) (* -0.0003968253968253968 (pow im 7.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 480.0) {
		tmp = 0.5 * (cos(re) * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0))));
	} else if (im <= 1.1e+44) {
		tmp = log1p(expm1(-im));
	} else {
		tmp = 0.5 * (cos(re) * (-0.0003968253968253968 * pow(im, 7.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 480.0) {
		tmp = 0.5 * (Math.cos(re) * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0))));
	} else if (im <= 1.1e+44) {
		tmp = Math.log1p(Math.expm1(-im));
	} else {
		tmp = 0.5 * (Math.cos(re) * (-0.0003968253968253968 * Math.pow(im, 7.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 480.0:
		tmp = 0.5 * (math.cos(re) * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0))))
	elif im <= 1.1e+44:
		tmp = math.log1p(math.expm1(-im))
	else:
		tmp = 0.5 * (math.cos(re) * (-0.0003968253968253968 * math.pow(im, 7.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 480.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0)))));
	elseif (im <= 1.1e+44)
		tmp = log1p(expm1(Float64(-im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-0.0003968253968253968 * (im ^ 7.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 480.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 480

   1. Initial program 36.0%

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

      1. cos-neg 36.0%

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

      2. sub-neg 36.0%

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

      3. neg-sub0 36.0%

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

      4. remove-double-neg 36.0%

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

      5. remove-double-neg 36.0%

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

      6. sub0-neg 36.0%

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

      7. distribute-neg-in 36.0%

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

      8. +-commutative 36.0%

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

      9. sub-neg 36.0%

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

      10. associate-*l* 36.0%

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

      11. sub-neg 36.0%

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

      12. +-commutative 36.0%

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

      13. distribute-neg-in 36.0%

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

   3. Simplified 36.0%

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

   5. Taylor expanded in im around 0 92.6%

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

   if 480 < im < 1.09999999999999998e44

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

      1. log1p-expm1-u 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 80.0%

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

      1. expm1-def 80.0%

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

      2. mul-1-neg 80.0%

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

   10. Simplified 80.0%

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

   if 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.0003968253968253968 \cdot {im}^{7}\right) \cdot \cos re\right)} \]

      2. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)} \]

   8. Simplified 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 480:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.7:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+236}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{elif}\;im \leq 3.98 \cdot 10^{+251}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.7)
   (* (- im) (cos re))
   (if (<= im 6e+236)
     (log1p (expm1 (- im)))
     (if (<= im 3.98e+251)
       (* 0.5 (* im (fma re re -2.0)))
       (* 0.5 (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.7) {
		tmp = -im * cos(re);
	} else if (im <= 6e+236) {
		tmp = log1p(expm1(-im));
	} else if (im <= 3.98e+251) {
		tmp = 0.5 * (im * fma(re, re, -2.0));
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.7)
		tmp = Float64(Float64(-im) * cos(re));
	elseif (im <= 6e+236)
		tmp = log1p(expm1(Float64(-im)));
	elseif (im <= 3.98e+251)
		tmp = Float64(0.5 * Float64(im * fma(re, re, -2.0)));
	else
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.7], N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6e+236], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 3.98e+251], N[(0.5 * N[(im * N[(re * re + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.69999999999999996

   1. Initial program 36.0%

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

      1. cos-neg 36.0%

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

      2. sub-neg 36.0%

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

      3. neg-sub0 36.0%

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

      4. remove-double-neg 36.0%

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

      5. remove-double-neg 36.0%

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

      6. sub0-neg 36.0%

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

      7. distribute-neg-in 36.0%

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

      8. +-commutative 36.0%

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

      9. sub-neg 36.0%

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

      10. associate-*l* 36.0%

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

      11. sub-neg 36.0%

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

      12. +-commutative 36.0%

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

      13. distribute-neg-in 36.0%

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

   3. Simplified 36.0%

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

   5. Taylor expanded in im around 0 70.9%

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

   6. Taylor expanded in im around 0 70.9%

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

      1. associate-*r* 70.9%

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

      2. *-commutative 70.9%

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

      3. mul-1-neg 70.9%

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

   8. Simplified 70.9%

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

   if 1.69999999999999996 < im < 5.9999999999999996e236

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 4.5%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im\right)} \cdot \cos re\right) \]
   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 80.4%

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

      1. expm1-def 80.4%

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

      2. mul-1-neg 80.4%

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

   10. Simplified 80.4%

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

   if 5.9999999999999996e236 < im < 3.98e251

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 6.7%

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

   6. Taylor expanded in re around 0 0.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right) + im \cdot {re}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot {re}^{2} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)}\right) \]

      2. associate-+r+ 0.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-2 \cdot im + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]

      3. *-commutative 0.0%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{im \cdot -2} + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      4. distribute-lft-out 0.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot \left(-2 + {re}^{2}\right)} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      5. *-commutative 0.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(im \cdot {re}^{4}\right) \cdot -0.08333333333333333}\right) \]

      6. *-commutative 0.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left({re}^{4} \cdot im\right)} \cdot -0.08333333333333333\right) \]

      7. associate-*l* 0.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{{re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)}\right) \]

   8. Simplified 0.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 + {re}^{2}\right) + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   9. Taylor expanded in re around 0 100.0%

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

       1. *-commutative 100.0%

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

       2. distribute-rgt-in 100.0%

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

       3. +-commutative 100.0%

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

       4. unpow2 100.0%

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

       5. fma-udef 100.0%

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

   11. Simplified 100.0%

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

   if 3.98e251 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 80.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.7:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+236}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{elif}\;im \leq 3.98 \cdot 10^{+251}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 6.7 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+236} \lor \neg \left(im \leq 3.98 \cdot 10^{+251}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 720.0)
   (* (- im) (cos re))
   (if (<= im 6.7e+100)
     (* 0.5 (* (pow re 4.0) (* im -0.08333333333333333)))
     (if (or (<= im 6e+236) (not (<= im 3.98e+251)))
       (* 0.5 (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0))))
       (* 0.5 (* im (fma re re -2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = -im * cos(re);
	} else if (im <= 6.7e+100) {
		tmp = 0.5 * (pow(re, 4.0) * (im * -0.08333333333333333));
	} else if ((im <= 6e+236) || !(im <= 3.98e+251)) {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0)));
	} else {
		tmp = 0.5 * (im * fma(re, re, -2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 720.0)
		tmp = Float64(Float64(-im) * cos(re));
	elseif (im <= 6.7e+100)
		tmp = Float64(0.5 * Float64((re ^ 4.0) * Float64(im * -0.08333333333333333)));
	elseif ((im <= 6e+236) || !(im <= 3.98e+251))
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0))));
	else
		tmp = Float64(0.5 * Float64(im * fma(re, re, -2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 720.0], N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.7e+100], N[(0.5 * N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 6e+236], N[Not[LessEqual[im, 3.98e+251]], $MachinePrecision]], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(re * re + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 720

   1. Initial program 36.0%

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

      1. cos-neg 36.0%

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

      2. sub-neg 36.0%

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

      3. neg-sub0 36.0%

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

      4. remove-double-neg 36.0%

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

      5. remove-double-neg 36.0%

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

      6. sub0-neg 36.0%

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

      7. distribute-neg-in 36.0%

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

      8. +-commutative 36.0%

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

      9. sub-neg 36.0%

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

      10. associate-*l* 36.0%

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

      11. sub-neg 36.0%

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

      12. +-commutative 36.0%

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

      13. distribute-neg-in 36.0%

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

   3. Simplified 36.0%

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

   5. Taylor expanded in im around 0 70.9%

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

   6. Taylor expanded in im around 0 70.9%

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

      1. associate-*r* 70.9%

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

      2. *-commutative 70.9%

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

      3. mul-1-neg 70.9%

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

   8. Simplified 70.9%

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

   if 720 < im < 6.6999999999999997e100

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.5%

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

   6. Taylor expanded in re around 0 12.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right) + im \cdot {re}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 12.7%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot {re}^{2} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)}\right) \]

      2. associate-+r+ 12.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-2 \cdot im + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]

      3. *-commutative 12.7%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{im \cdot -2} + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      4. distribute-lft-out 12.7%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot \left(-2 + {re}^{2}\right)} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      5. *-commutative 12.7%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(im \cdot {re}^{4}\right) \cdot -0.08333333333333333}\right) \]

      6. *-commutative 12.7%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left({re}^{4} \cdot im\right)} \cdot -0.08333333333333333\right) \]

      7. associate-*l* 12.7%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{{re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)}\right) \]

   8. Simplified 12.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 + {re}^{2}\right) + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   9. Taylor expanded in re around inf 27.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 27.8%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{4}\right)} \]

       2. *-commutative 27.8%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot {re}^{4}\right) \]

       3. *-commutative 27.8%

          \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   11. Simplified 27.8%

       \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   if 6.6999999999999997e100 < im < 5.9999999999999996e236 or 3.98e251 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 96.6%

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

   6. Taylor expanded in re around 0 77.4%

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

   if 5.9999999999999996e236 < im < 3.98e251

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 6.7%

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

   6. Taylor expanded in re around 0 0.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right) + im \cdot {re}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot {re}^{2} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)}\right) \]

      2. associate-+r+ 0.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-2 \cdot im + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]

      3. *-commutative 0.0%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{im \cdot -2} + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      4. distribute-lft-out 0.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot \left(-2 + {re}^{2}\right)} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      5. *-commutative 0.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(im \cdot {re}^{4}\right) \cdot -0.08333333333333333}\right) \]

      6. *-commutative 0.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left({re}^{4} \cdot im\right)} \cdot -0.08333333333333333\right) \]

      7. associate-*l* 0.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{{re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)}\right) \]

   8. Simplified 0.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 + {re}^{2}\right) + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   9. Taylor expanded in re around 0 100.0%

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

       1. *-commutative 100.0%

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

       2. distribute-rgt-in 100.0%

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

       3. +-commutative 100.0%

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

       4. unpow2 100.0%

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

       5. fma-udef 100.0%

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

   11. Simplified 100.0%

       \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 6.7 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+236} \lor \neg \left(im \leq 3.98 \cdot 10^{+251}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{+119} \lor \neg \left(im \leq 9.2 \cdot 10^{+260}\right) \land im \leq 4.6 \cdot 10^{+293}:\\ \;\;\;\;0.5 \cdot \left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 700.0)
   (* (- im) (cos re))
   (if (or (<= im 2.05e+119) (and (not (<= im 9.2e+260)) (<= im 4.6e+293)))
     (* 0.5 (* (pow re 4.0) (* im -0.08333333333333333)))
     (* 0.5 (* im (fma re re -2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 700.0) {
		tmp = -im * cos(re);
	} else if ((im <= 2.05e+119) || (!(im <= 9.2e+260) && (im <= 4.6e+293))) {
		tmp = 0.5 * (pow(re, 4.0) * (im * -0.08333333333333333));
	} else {
		tmp = 0.5 * (im * fma(re, re, -2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 700.0)
		tmp = Float64(Float64(-im) * cos(re));
	elseif ((im <= 2.05e+119) || (!(im <= 9.2e+260) && (im <= 4.6e+293)))
		tmp = Float64(0.5 * Float64((re ^ 4.0) * Float64(im * -0.08333333333333333)));
	else
		tmp = Float64(0.5 * Float64(im * fma(re, re, -2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 700.0], N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 2.05e+119], And[N[Not[LessEqual[im, 9.2e+260]], $MachinePrecision], LessEqual[im, 4.6e+293]]], N[(0.5 * N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(re * re + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 700

   1. Initial program 36.0%

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

      1. cos-neg 36.0%

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

      2. sub-neg 36.0%

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

      3. neg-sub0 36.0%

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

      4. remove-double-neg 36.0%

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

      5. remove-double-neg 36.0%

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

      6. sub0-neg 36.0%

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

      7. distribute-neg-in 36.0%

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

      8. +-commutative 36.0%

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

      9. sub-neg 36.0%

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

      10. associate-*l* 36.0%

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

      11. sub-neg 36.0%

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

      12. +-commutative 36.0%

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

      13. distribute-neg-in 36.0%

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

   3. Simplified 36.0%

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

   5. Taylor expanded in im around 0 70.9%

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

   6. Taylor expanded in im around 0 70.9%

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

      1. associate-*r* 70.9%

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

      2. *-commutative 70.9%

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

      3. mul-1-neg 70.9%

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

   8. Simplified 70.9%

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

   if 700 < im < 2.0499999999999999e119 or 9.20000000000000044e260 < im < 4.5999999999999998e293

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 5.2%

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

   6. Taylor expanded in re around 0 11.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right) + im \cdot {re}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 11.4%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot {re}^{2} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)}\right) \]

      2. associate-+r+ 11.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-2 \cdot im + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]

      3. *-commutative 11.4%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{im \cdot -2} + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      4. distribute-lft-out 11.4%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot \left(-2 + {re}^{2}\right)} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      5. *-commutative 11.4%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(im \cdot {re}^{4}\right) \cdot -0.08333333333333333}\right) \]

      6. *-commutative 11.4%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left({re}^{4} \cdot im\right)} \cdot -0.08333333333333333\right) \]

      7. associate-*l* 11.4%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{{re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)}\right) \]

   8. Simplified 11.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 + {re}^{2}\right) + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   9. Taylor expanded in re around inf 34.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 34.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{4}\right)} \]

       2. *-commutative 34.7%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot {re}^{4}\right) \]

       3. *-commutative 34.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   11. Simplified 34.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   if 2.0499999999999999e119 < im < 9.20000000000000044e260 or 4.5999999999999998e293 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 8.3%

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

   6. Taylor expanded in re around 0 10.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right) + im \cdot {re}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 10.6%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot {re}^{2} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)}\right) \]

      2. associate-+r+ 10.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-2 \cdot im + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]

      3. *-commutative 10.6%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{im \cdot -2} + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      4. distribute-lft-out 10.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot \left(-2 + {re}^{2}\right)} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      5. *-commutative 10.6%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(im \cdot {re}^{4}\right) \cdot -0.08333333333333333}\right) \]

      6. *-commutative 10.6%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left({re}^{4} \cdot im\right)} \cdot -0.08333333333333333\right) \]

      7. associate-*l* 10.6%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{{re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)}\right) \]

   8. Simplified 10.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 + {re}^{2}\right) + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   9. Taylor expanded in re around 0 35.0%

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

       1. *-commutative 35.0%

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

       2. distribute-rgt-in 35.0%

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

       3. +-commutative 35.0%

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

       4. unpow2 35.0%

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

       5. fma-udef 35.0%

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

   11. Simplified 35.0%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{+119} \lor \neg \left(im \leq 9.2 \cdot 10^{+260}\right) \land im \leq 4.6 \cdot 10^{+293}:\\ \;\;\;\;0.5 \cdot \left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.3 \cdot 10^{+39}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.3e+39) (* (- im) (cos re)) (* 0.5 (* im (fma re re -2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.3e+39) {
		tmp = -im * cos(re);
	} else {
		tmp = 0.5 * (im * fma(re, re, -2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.3e+39)
		tmp = Float64(Float64(-im) * cos(re));
	else
		tmp = Float64(0.5 * Float64(im * fma(re, re, -2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.3e+39], N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(re * re + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.3e39

   1. Initial program 37.7%

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

      1. cos-neg 37.7%

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

      2. sub-neg 37.7%

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

      3. neg-sub0 37.7%

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

      4. remove-double-neg 37.7%

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

      5. remove-double-neg 37.7%

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

      6. sub0-neg 37.7%

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

      7. distribute-neg-in 37.7%

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

      8. +-commutative 37.7%

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

      9. sub-neg 37.7%

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

      10. associate-*l* 37.7%

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

      11. sub-neg 37.7%

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

      12. +-commutative 37.7%

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

      13. distribute-neg-in 37.7%

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

   3. Simplified 37.7%

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

   5. Taylor expanded in im around 0 69.1%

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

   6. Taylor expanded in im around 0 69.1%

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

      1. associate-*r* 69.1%

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

      2. *-commutative 69.1%

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

      3. mul-1-neg 69.1%

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

   8. Simplified 69.1%

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

   if 4.3e39 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 7.0%

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

   6. Taylor expanded in re around 0 10.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right) + im \cdot {re}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 10.1%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot {re}^{2} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)}\right) \]

      2. associate-+r+ 10.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-2 \cdot im + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]

      3. *-commutative 10.1%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{im \cdot -2} + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      4. distribute-lft-out 10.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot \left(-2 + {re}^{2}\right)} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      5. *-commutative 10.1%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(im \cdot {re}^{4}\right) \cdot -0.08333333333333333}\right) \]

      6. *-commutative 10.1%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left({re}^{4} \cdot im\right)} \cdot -0.08333333333333333\right) \]

      7. associate-*l* 10.1%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{{re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)}\right) \]

   8. Simplified 10.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 + {re}^{2}\right) + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   9. Taylor expanded in re around 0 23.3%

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

       1. *-commutative 23.3%

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

       2. distribute-rgt-in 23.3%

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

       3. +-commutative 23.3%

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

       4. unpow2 23.3%

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

       5. fma-udef 23.3%

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

   11. Simplified 23.3%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.3 \cdot 10^{+39}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \mathsf{fma}\left(re, re, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+46}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+46) (* (- im) (cos re)) (* 0.5 (* im (pow re 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+46) {
		tmp = -im * cos(re);
	} else {
		tmp = 0.5 * (im * pow(re, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.1d+46) then
        tmp = -im * cos(re)
    else
        tmp = 0.5d0 * (im * (re ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+46) {
		tmp = -im * Math.cos(re);
	} else {
		tmp = 0.5 * (im * Math.pow(re, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+46:
		tmp = -im * math.cos(re)
	else:
		tmp = 0.5 * (im * math.pow(re, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+46)
		tmp = Float64(Float64(-im) * cos(re));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+46)
		tmp = -im * cos(re);
	else
		tmp = 0.5 * (im * (re ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+46], N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.1e46

   1. Initial program 37.7%

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

      1. cos-neg 37.7%

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

      2. sub-neg 37.7%

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

      3. neg-sub0 37.7%

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

      4. remove-double-neg 37.7%

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

      5. remove-double-neg 37.7%

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

      6. sub0-neg 37.7%

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

      7. distribute-neg-in 37.7%

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

      8. +-commutative 37.7%

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

      9. sub-neg 37.7%

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

      10. associate-*l* 37.7%

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

      11. sub-neg 37.7%

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

      12. +-commutative 37.7%

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

      13. distribute-neg-in 37.7%

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

   3. Simplified 37.7%

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

   5. Taylor expanded in im around 0 69.1%

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

   6. Taylor expanded in im around 0 69.1%

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

      1. associate-*r* 69.1%

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

      2. *-commutative 69.1%

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

      3. mul-1-neg 69.1%

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

   8. Simplified 69.1%

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

   if 2.1e46 < im

   1. Initial program 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 7.0%

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

   6. Taylor expanded in re around 0 10.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right) + im \cdot {re}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative 10.1%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot {re}^{2} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)}\right) \]

      2. associate-+r+ 10.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-2 \cdot im + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]

      3. *-commutative 10.1%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{im \cdot -2} + im \cdot {re}^{2}\right) + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      4. distribute-lft-out 10.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot \left(-2 + {re}^{2}\right)} + -0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right) \]

      5. *-commutative 10.1%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(im \cdot {re}^{4}\right) \cdot -0.08333333333333333}\right) \]

      6. *-commutative 10.1%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left({re}^{4} \cdot im\right)} \cdot -0.08333333333333333\right) \]

      7. associate-*l* 10.1%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{{re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)}\right) \]

   8. Simplified 10.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 + {re}^{2}\right) + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   9. Taylor expanded in re around 0 23.3%

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

       1. *-commutative 23.3%

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

       2. distribute-rgt-in 23.3%

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

       3. +-commutative 23.3%

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

       4. unpow2 23.3%

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

       5. fma-udef 23.3%

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

   11. Simplified 23.3%

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

   12. Taylor expanded in re around inf 19.8%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+46}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. cos-neg 54.2%

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

   2. sub-neg 54.2%

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

   3. neg-sub0 54.2%

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

   4. remove-double-neg 54.2%

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

   5. remove-double-neg 54.2%

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

   6. sub0-neg 54.2%

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

   7. distribute-neg-in 54.2%

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

   8. +-commutative 54.2%

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

   9. sub-neg 54.2%

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

   10. associate-*l* 54.2%

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

   11. sub-neg 54.2%

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

   12. +-commutative 54.2%

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

   13. distribute-neg-in 54.2%

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

3. Simplified 54.2%

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

5. Taylor expanded in im around 0 52.6%

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

6. Taylor expanded in im around 0 52.3%

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

   1. associate-*r* 52.3%

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

   2. *-commutative 52.3%

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

   3. mul-1-neg 52.3%

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

8. Simplified 52.3%

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

9. Final simplification 52.3%

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

Alternative 10: 30.5% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. cos-neg 54.2%

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

   2. sub-neg 54.2%

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

   3. neg-sub0 54.2%

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

   4. remove-double-neg 54.2%

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

   5. remove-double-neg 54.2%

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

   6. sub0-neg 54.2%

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

   7. distribute-neg-in 54.2%

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

   8. +-commutative 54.2%

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

   9. sub-neg 54.2%

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

   10. associate-*l* 54.2%

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

   11. sub-neg 54.2%

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

   12. +-commutative 54.2%

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

   13. distribute-neg-in 54.2%

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

3. Simplified 54.2%

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

5. Taylor expanded in im around 0 52.6%

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

6. Taylor expanded in re around 0 30.0%

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

7. Final simplification 30.0%

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

Alternative 11: 30.5% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. cos-neg 54.2%

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

   2. sub-neg 54.2%

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

   3. neg-sub0 54.2%

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

   4. remove-double-neg 54.2%

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

   5. remove-double-neg 54.2%

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

   6. sub0-neg 54.2%

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

   7. distribute-neg-in 54.2%

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

   8. +-commutative 54.2%

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

   9. sub-neg 54.2%

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

   10. associate-*l* 54.2%

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

   11. sub-neg 54.2%

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

   12. +-commutative 54.2%

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

   13. distribute-neg-in 54.2%

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

3. Simplified 54.2%

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

5. Taylor expanded in im around 0 52.6%

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

6. Taylor expanded in re around 0 29.7%

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

   1. mul-1-neg 29.7%

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

8. Simplified 29.7%

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

9. Final simplification 29.7%

   \[\leadsto -im \]
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 2024031 
(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))))