math.sin on complex, imaginary part

Percentage Accurate: 53.5% → 99.1%

Time: 17.8s

Alternatives: 7

Speedup: 2.6×

• 49.1% 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: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.5%

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

   1. /-rgt-identity 58.5%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 58.5%

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

   3. associate-*l/ 58.5%

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

   4. cos-neg 58.5%

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

   5. associate-*l* 58.5%

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

   6. associate-*r/ 58.5%

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

   7. exp-0 58.5%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 58.5%

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

   9. *-commutative 58.5%

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

   10. neg-sub0 58.5%

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

   11. cos-neg 58.5%

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

3. Simplified 58.5%

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

5. Taylor expanded in im around 0 47.5%

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

   1. log1p-expm1-u 98.8%

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

   2. *-commutative 98.8%

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

   3. associate-*l* 98.8%

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

7. Applied egg-rr 98.8%

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

Alternative 2: 91.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 430:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot {im}^{7}\right) \cdot -0.0001984126984126984\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 430.0)
   (* 0.5 (* (cos re) (* im (- (* -0.3333333333333333 (pow im 2.0)) 2.0))))
   (if (<= im 4.1e+41)
     (* 0.5 (log1p (expm1 (* im -2.0))))
     (* (* (cos re) (pow im 7.0)) -0.0001984126984126984))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 430.0) {
		tmp = 0.5 * (cos(re) * (im * ((-0.3333333333333333 * pow(im, 2.0)) - 2.0)));
	} else if (im <= 4.1e+41) {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	} else {
		tmp = (cos(re) * pow(im, 7.0)) * -0.0001984126984126984;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 430.0) {
		tmp = 0.5 * (Math.cos(re) * (im * ((-0.3333333333333333 * Math.pow(im, 2.0)) - 2.0)));
	} else if (im <= 4.1e+41) {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	} else {
		tmp = (Math.cos(re) * Math.pow(im, 7.0)) * -0.0001984126984126984;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 430.0:
		tmp = 0.5 * (math.cos(re) * (im * ((-0.3333333333333333 * math.pow(im, 2.0)) - 2.0)))
	elif im <= 4.1e+41:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	else:
		tmp = (math.cos(re) * math.pow(im, 7.0)) * -0.0001984126984126984
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 430.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * Float64(Float64(-0.3333333333333333 * (im ^ 2.0)) - 2.0))));
	elseif (im <= 4.1e+41)
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	else
		tmp = Float64(Float64(cos(re) * (im ^ 7.0)) * -0.0001984126984126984);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 430.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * N[(N[(-0.3333333333333333 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.1e+41], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 430

   1. Initial program 44.4%

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

      1. /-rgt-identity 44.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 44.4%

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

      3. associate-*l/ 44.4%

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

      4. cos-neg 44.4%

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

      5. associate-*l* 44.4%

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

      6. associate-*r/ 44.4%

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

      7. exp-0 44.4%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 44.4%

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

      9. *-commutative 44.4%

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

      10. neg-sub0 44.4%

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

      11. cos-neg 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in im around 0 93.0%

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

   if 430 < im < 4.1000000000000004e41

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.4%

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

      1. log1p-expm1-u 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-*l* 89.5%

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

   7. Applied egg-rr 89.5%

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

   8. Taylor expanded in re around 0 78.4%

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

      1. expm1-define 78.4%

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

   10. Simplified 78.4%

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

   if 4.1000000000000004e41 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   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 96.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 96.9%

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

      2. associate-+r+ 96.9%

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

      3. associate-*r* 96.9%

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

      4. *-commutative 96.9%

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

      5. distribute-rgt-out 96.9%

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

      6. +-commutative 96.9%

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

      7. metadata-eval 96.9%

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

      8. sub-neg 96.9%

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

      9. fma-define 96.9%

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

   7. Simplified 96.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(\cos re, \mathsf{fma}\left(-0.3333333333333333, {im}^{2}, -2\right), \cos re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0003968253968253968, -0.016666666666666666\right) \cdot {im}^{4}\right)\right)\right)} \]

   8. Taylor expanded in im around inf 98.4%

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

      1. associate-*r* 98.4%

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

      2. *-commutative 98.4%

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

   10. Simplified 98.4%

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

   11. Taylor expanded in re around inf 98.4%

       \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]
   12. Step-by-step derivation

       1. *-commutative 98.4%

          \[\leadsto \color{blue}{\left({im}^{7} \cdot \cos re\right) \cdot -0.0001984126984126984} \]

       2. *-commutative 98.4%

          \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{7}\right)} \cdot -0.0001984126984126984 \]

   13. Simplified 98.4%

       \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 430:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot {im}^{7}\right) \cdot -0.0001984126984126984\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 8.4 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot {im}^{7}\right) \cdot -0.0001984126984126984\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.6e-15)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 8.4e+18)
     (* 0.5 (log1p (expm1 (* im -2.0))))
     (* (* (cos re) (pow im 7.0)) -0.0001984126984126984))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.6e-15) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 8.4e+18) {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	} else {
		tmp = (cos(re) * pow(im, 7.0)) * -0.0001984126984126984;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.6e-15) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 8.4e+18) {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	} else {
		tmp = (Math.cos(re) * Math.pow(im, 7.0)) * -0.0001984126984126984;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.6e-15:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 8.4e+18:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	else:
		tmp = (math.cos(re) * math.pow(im, 7.0)) * -0.0001984126984126984
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.6e-15)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 8.4e+18)
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	else
		tmp = Float64(Float64(cos(re) * (im ^ 7.0)) * -0.0001984126984126984);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.6e-15], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.4e+18], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.60000000000000004e-15

   1. Initial program 44.2%

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

      1. /-rgt-identity 44.2%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 44.2%

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

      3. associate-*l/ 44.2%

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

      4. cos-neg 44.2%

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

      5. associate-*l* 44.2%

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

      6. associate-*r/ 44.2%

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

      7. exp-0 44.2%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 44.2%

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

      9. *-commutative 44.2%

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

      10. neg-sub0 44.2%

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

      11. cos-neg 44.2%

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

   3. Simplified 44.2%

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

   5. Taylor expanded in im around 0 61.9%

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

   if 2.60000000000000004e-15 < im < 8.4e18

   1. Initial program 89.2%

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

      1. /-rgt-identity 89.2%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 89.2%

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

      3. associate-*l/ 89.2%

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

      4. cos-neg 89.2%

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

      5. associate-*l* 89.2%

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

      6. associate-*r/ 89.2%

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

      7. exp-0 89.2%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 89.2%

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

      9. *-commutative 89.2%

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

      10. neg-sub0 89.2%

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

      11. cos-neg 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in im around 0 21.6%

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

      1. log1p-expm1-u 77.2%

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

      2. *-commutative 77.2%

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

      3. associate-*l* 77.2%

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

   7. Applied egg-rr 77.2%

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

   8. Taylor expanded in re around 0 62.2%

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

      1. expm1-define 72.5%

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

   10. Simplified 72.5%

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

   if 8.4e18 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   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 90.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 90.9%

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

      2. associate-+r+ 90.9%

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

      3. associate-*r* 90.9%

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

      4. *-commutative 90.9%

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

      5. distribute-rgt-out 90.9%

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

      6. +-commutative 90.9%

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

      7. metadata-eval 90.9%

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

      8. sub-neg 90.9%

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

      9. fma-define 90.9%

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

   7. Simplified 90.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(\cos re, \mathsf{fma}\left(-0.3333333333333333, {im}^{2}, -2\right), \cos re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0003968253968253968, -0.016666666666666666\right) \cdot {im}^{4}\right)\right)\right)} \]

   8. Taylor expanded in im around inf 92.2%

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

      1. associate-*r* 92.2%

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

      2. *-commutative 92.2%

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

   10. Simplified 92.2%

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

   11. Taylor expanded in re around inf 92.2%

       \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]
   12. Step-by-step derivation

       1. *-commutative 92.2%

          \[\leadsto \color{blue}{\left({im}^{7} \cdot \cos re\right) \cdot -0.0001984126984126984} \]

       2. *-commutative 92.2%

          \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{7}\right)} \cdot -0.0001984126984126984 \]

   13. Simplified 92.2%

       \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 8.4 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot {im}^{7}\right) \cdot -0.0001984126984126984\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{7} \cdot 0.0001984126984126984\\ t_1 := {im}^{7} \cdot -0.0001984126984126984\\ \mathbf{if}\;im \leq 0.2:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;im \leq 1.72 \cdot 10^{+162}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(0.26666666666666666 \cdot {im}^{4} - 2\right)\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+178} \lor \neg \left(im \leq 3.3 \cdot 10^{+184}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 7.0) 0.0001984126984126984))
        (t_1 (* (pow im 7.0) -0.0001984126984126984)))
   (if (<= im 0.2)
     (* 0.5 (* (cos re) (* im -2.0)))
     (if (<= im 6.5e+93)
       (* 0.5 (log1p (expm1 (* im -2.0))))
       (if (<= im 6.6e+93)
         t_0
         (if (<= im 3.6e+151)
           t_1
           (if (<= im 1.72e+162)
             (* 0.5 (* im (- (* 0.26666666666666666 (pow im 4.0)) 2.0)))
             (if (or (<= im 2.1e+178) (not (<= im 3.3e+184))) t_1 t_0))))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 7.0) * 0.0001984126984126984;
	double t_1 = pow(im, 7.0) * -0.0001984126984126984;
	double tmp;
	if (im <= 0.2) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 6.5e+93) {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	} else if (im <= 6.6e+93) {
		tmp = t_0;
	} else if (im <= 3.6e+151) {
		tmp = t_1;
	} else if (im <= 1.72e+162) {
		tmp = 0.5 * (im * ((0.26666666666666666 * pow(im, 4.0)) - 2.0));
	} else if ((im <= 2.1e+178) || !(im <= 3.3e+184)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 7.0) * 0.0001984126984126984;
	double t_1 = Math.pow(im, 7.0) * -0.0001984126984126984;
	double tmp;
	if (im <= 0.2) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 6.5e+93) {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	} else if (im <= 6.6e+93) {
		tmp = t_0;
	} else if (im <= 3.6e+151) {
		tmp = t_1;
	} else if (im <= 1.72e+162) {
		tmp = 0.5 * (im * ((0.26666666666666666 * Math.pow(im, 4.0)) - 2.0));
	} else if ((im <= 2.1e+178) || !(im <= 3.3e+184)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 7.0) * 0.0001984126984126984
	t_1 = math.pow(im, 7.0) * -0.0001984126984126984
	tmp = 0
	if im <= 0.2:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 6.5e+93:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	elif im <= 6.6e+93:
		tmp = t_0
	elif im <= 3.6e+151:
		tmp = t_1
	elif im <= 1.72e+162:
		tmp = 0.5 * (im * ((0.26666666666666666 * math.pow(im, 4.0)) - 2.0))
	elif (im <= 2.1e+178) or not (im <= 3.3e+184):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 7.0) * 0.0001984126984126984)
	t_1 = Float64((im ^ 7.0) * -0.0001984126984126984)
	tmp = 0.0
	if (im <= 0.2)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 6.5e+93)
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	elseif (im <= 6.6e+93)
		tmp = t_0;
	elseif (im <= 3.6e+151)
		tmp = t_1;
	elseif (im <= 1.72e+162)
		tmp = Float64(0.5 * Float64(im * Float64(Float64(0.26666666666666666 * (im ^ 4.0)) - 2.0)));
	elseif ((im <= 2.1e+178) || !(im <= 3.3e+184))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 7.0], $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]}, If[LessEqual[im, 0.2], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.5e+93], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.6e+93], t$95$0, If[LessEqual[im, 3.6e+151], t$95$1, If[LessEqual[im, 1.72e+162], N[(0.5 * N[(im * N[(N[(0.26666666666666666 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 2.1e+178], N[Not[LessEqual[im, 3.3e+184]], $MachinePrecision]], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 0.20000000000000001

   1. Initial program 44.4%

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

      1. /-rgt-identity 44.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 44.4%

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

      3. associate-*l/ 44.4%

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

      4. cos-neg 44.4%

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

      5. associate-*l* 44.4%

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

      6. associate-*r/ 44.4%

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

      7. exp-0 44.4%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 44.4%

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

      9. *-commutative 44.4%

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

      10. neg-sub0 44.4%

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

      11. cos-neg 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in im around 0 61.9%

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

   if 0.20000000000000001 < im < 6.4999999999999998e93

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.5%

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

      1. log1p-expm1-u 94.8%

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

      2. *-commutative 94.8%

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

      3. associate-*l* 94.8%

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

   7. Applied egg-rr 94.8%

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

   8. Taylor expanded in re around 0 78.1%

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

      1. expm1-define 78.1%

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

   10. Simplified 78.1%

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

   if 6.4999999999999998e93 < im < 6.60000000000000017e93 or 2.0999999999999999e178 < im < 3.2999999999999998e184

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   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%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. associate-+r+ 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. +-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. sub-neg 100.0%

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

      9. fma-define 100.0%

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

   7. Simplified 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(\cos re, \mathsf{fma}\left(-0.3333333333333333, {im}^{2}, -2\right), \cos re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0003968253968253968, -0.016666666666666666\right) \cdot {im}^{4}\right)\right)\right)} \]

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]
   9. 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%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in re around 0 0.0%

       \[\leadsto \color{blue}{-0.0001984126984126984 \cdot {im}^{7}} \]
   12. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{{im}^{7} \cdot -0.0001984126984126984} \]

   13. Simplified 0.0%

       \[\leadsto \color{blue}{{im}^{7} \cdot -0.0001984126984126984} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \color{blue}{\sqrt{{im}^{7} \cdot -0.0001984126984126984} \cdot \sqrt{{im}^{7} \cdot -0.0001984126984126984}} \]

       2. sqrt-unprod 100.0%

          \[\leadsto \color{blue}{\sqrt{\left({im}^{7} \cdot -0.0001984126984126984\right) \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)}} \]

       3. swap-sqr 100.0%

          \[\leadsto \sqrt{\color{blue}{\left({im}^{7} \cdot {im}^{7}\right) \cdot \left(-0.0001984126984126984 \cdot -0.0001984126984126984\right)}} \]

       4. pow-prod-up 100.0%

          \[\leadsto \sqrt{\color{blue}{{im}^{\left(7 + 7\right)}} \cdot \left(-0.0001984126984126984 \cdot -0.0001984126984126984\right)} \]

       5. metadata-eval 100.0%

          \[\leadsto \sqrt{{im}^{\color{blue}{14}} \cdot \left(-0.0001984126984126984 \cdot -0.0001984126984126984\right)} \]

       6. metadata-eval 100.0%

          \[\leadsto \sqrt{{im}^{14} \cdot \color{blue}{3.936759889140842 \cdot 10^{-8}}} \]

   15. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\sqrt{{im}^{14} \cdot 3.936759889140842 \cdot 10^{-8}}} \]
   16. Step-by-step derivation

       1. sqrt-prod 100.0%

          \[\leadsto \color{blue}{\sqrt{{im}^{14}} \cdot \sqrt{3.936759889140842 \cdot 10^{-8}}} \]

       2. sqrt-pow1 100.0%

          \[\leadsto \color{blue}{{im}^{\left(\frac{14}{2}\right)}} \cdot \sqrt{3.936759889140842 \cdot 10^{-8}} \]

       3. metadata-eval 100.0%

          \[\leadsto {im}^{\color{blue}{7}} \cdot \sqrt{3.936759889140842 \cdot 10^{-8}} \]

       4. metadata-eval 100.0%

          \[\leadsto {im}^{7} \cdot \color{blue}{0.0001984126984126984} \]

   17. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{{im}^{7} \cdot 0.0001984126984126984} \]

   if 6.60000000000000017e93 < im < 3.6e151 or 1.72e162 < im < 2.0999999999999999e178 or 3.2999999999999998e184 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   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%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. associate-+r+ 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. +-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. sub-neg 100.0%

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

      9. fma-define 100.0%

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

   7. Simplified 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(\cos re, \mathsf{fma}\left(-0.3333333333333333, {im}^{2}, -2\right), \cos re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0003968253968253968, -0.016666666666666666\right) \cdot {im}^{4}\right)\right)\right)} \]

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]
   9. 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%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in re around 0 82.1%

       \[\leadsto \color{blue}{-0.0001984126984126984 \cdot {im}^{7}} \]
   12. Step-by-step derivation

       1. *-commutative 82.1%

          \[\leadsto \color{blue}{{im}^{7} \cdot -0.0001984126984126984} \]

   13. Simplified 82.1%

       \[\leadsto \color{blue}{{im}^{7} \cdot -0.0001984126984126984} \]

   if 3.6e151 < im < 1.72e162

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   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 4.7%

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

      1. log1p-expm1-u 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 0.0%

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

      1. expm1-define 0.0%

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

   10. Simplified 0.0%

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

   11. Taylor expanded in im around 0 100.0%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{im \cdot \left(im \cdot \left(2 + im \cdot \left(0.6666666666666666 \cdot im - 1.3333333333333333\right)\right) - 2\right)}\right) \]

   12. Taylor expanded in im around 0 100.0%

       \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(0.26666666666666666 \cdot {im}^{4} - 2\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.2:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+93}:\\ \;\;\;\;{im}^{7} \cdot 0.0001984126984126984\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+151}:\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{elif}\;im \leq 1.72 \cdot 10^{+162}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(0.26666666666666666 \cdot {im}^{4} - 2\right)\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+178} \lor \neg \left(im \leq 3.3 \cdot 10^{+184}\right):\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot 0.0001984126984126984\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+215} \lor \neg \left(im \leq 1.6 \cdot 10^{+227}\right):\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0.4444444444444444 \cdot {im}^{6}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.056)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (or (<= im 1.6e+215) (not (<= im 1.6e+227)))
     (* (pow im 7.0) -0.0001984126984126984)
     (* 0.5 (* 0.4444444444444444 (pow im 6.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if ((im <= 1.6e+215) || !(im <= 1.6e+227)) {
		tmp = pow(im, 7.0) * -0.0001984126984126984;
	} else {
		tmp = 0.5 * (0.4444444444444444 * pow(im, 6.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.056d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else if ((im <= 1.6d+215) .or. (.not. (im <= 1.6d+227))) then
        tmp = (im ** 7.0d0) * (-0.0001984126984126984d0)
    else
        tmp = 0.5d0 * (0.4444444444444444d0 * (im ** 6.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.056) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if ((im <= 1.6e+215) || !(im <= 1.6e+227)) {
		tmp = Math.pow(im, 7.0) * -0.0001984126984126984;
	} else {
		tmp = 0.5 * (0.4444444444444444 * Math.pow(im, 6.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.056:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif (im <= 1.6e+215) or not (im <= 1.6e+227):
		tmp = math.pow(im, 7.0) * -0.0001984126984126984
	else:
		tmp = 0.5 * (0.4444444444444444 * math.pow(im, 6.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.056)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif ((im <= 1.6e+215) || !(im <= 1.6e+227))
		tmp = Float64((im ^ 7.0) * -0.0001984126984126984);
	else
		tmp = Float64(0.5 * Float64(0.4444444444444444 * (im ^ 6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.056)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	elseif ((im <= 1.6e+215) || ~((im <= 1.6e+227)))
		tmp = (im ^ 7.0) * -0.0001984126984126984;
	else
		tmp = 0.5 * (0.4444444444444444 * (im ^ 6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.056], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 1.6e+215], N[Not[LessEqual[im, 1.6e+227]], $MachinePrecision]], N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision], N[(0.5 * N[(0.4444444444444444 * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0560000000000000012

   1. Initial program 44.1%

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

      1. /-rgt-identity 44.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 44.1%

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

      3. associate-*l/ 44.1%

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

      4. cos-neg 44.1%

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

      5. associate-*l* 44.1%

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

      6. associate-*r/ 44.1%

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

      7. exp-0 44.1%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 44.1%

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

      9. *-commutative 44.1%

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

      10. neg-sub0 44.1%

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

      11. cos-neg 44.1%

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

   3. Simplified 44.1%

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

   5. Taylor expanded in im around 0 62.1%

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

   if 0.0560000000000000012 < im < 1.5999999999999999e215 or 1.59999999999999994e227 < im

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 99.9%

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

      3. associate-*l/ 99.9%

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

      4. cos-neg 99.9%

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

      5. associate-*l* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. exp-0 99.9%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 99.9%

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

      9. *-commutative 99.9%

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

      10. neg-sub0 99.9%

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

      11. cos-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in im around 0 83.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 83.9%

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

      2. associate-+r+ 83.9%

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

      3. associate-*r* 83.9%

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

      4. *-commutative 83.9%

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

      5. distribute-rgt-out 83.9%

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

      6. +-commutative 83.9%

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

      7. metadata-eval 83.9%

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

      8. sub-neg 83.9%

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

      9. fma-define 83.9%

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

   7. Simplified 83.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(\cos re, \mathsf{fma}\left(-0.3333333333333333, {im}^{2}, -2\right), \cos re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0003968253968253968, -0.016666666666666666\right) \cdot {im}^{4}\right)\right)\right)} \]

   8. Taylor expanded in im around inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. *-commutative 83.8%

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

   10. Simplified 83.8%

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

   11. Taylor expanded in re around 0 60.0%

       \[\leadsto \color{blue}{-0.0001984126984126984 \cdot {im}^{7}} \]
   12. Step-by-step derivation

       1. *-commutative 60.0%

          \[\leadsto \color{blue}{{im}^{7} \cdot -0.0001984126984126984} \]

   13. Simplified 60.0%

       \[\leadsto \color{blue}{{im}^{7} \cdot -0.0001984126984126984} \]

   if 1.5999999999999999e215 < im < 1.59999999999999994e227

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   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 6.1%

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

      1. log1p-expm1-u 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 50.0%

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

      1. expm1-define 50.0%

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

   10. Simplified 50.0%

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

   11. Taylor expanded in im around 0 50.0%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{im \cdot \left(im \cdot \left(2 + im \cdot \left(0.6666666666666666 \cdot im - 1.3333333333333333\right)\right) - 2\right)}\right) \]

   12. Taylor expanded in im around 0 50.0%

       \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left({im}^{4} \cdot \left(0.26666666666666666 + 0.4444444444444444 \cdot im\right) - 2\right)\right)} \]

   13. Taylor expanded in im around inf 50.0%

       \[\leadsto 0.5 \cdot \color{blue}{\left(0.4444444444444444 \cdot {im}^{6}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.056:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{+215} \lor \neg \left(im \leq 1.6 \cdot 10^{+227}\right):\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0.4444444444444444 \cdot {im}^{6}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00078:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00078)
   (* 0.5 (* im -2.0))
   (* (pow im 7.0) -0.0001984126984126984)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00078) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = pow(im, 7.0) * -0.0001984126984126984;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00078d0) then
        tmp = 0.5d0 * (im * (-2.0d0))
    else
        tmp = (im ** 7.0d0) * (-0.0001984126984126984d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00078) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = Math.pow(im, 7.0) * -0.0001984126984126984;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00078:
		tmp = 0.5 * (im * -2.0)
	else:
		tmp = math.pow(im, 7.0) * -0.0001984126984126984
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00078)
		tmp = Float64(0.5 * Float64(im * -2.0));
	else
		tmp = Float64((im ^ 7.0) * -0.0001984126984126984);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00078)
		tmp = 0.5 * (im * -2.0);
	else
		tmp = (im ^ 7.0) * -0.0001984126984126984;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00078], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.79999999999999986e-4

   1. Initial program 44.1%

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

      1. /-rgt-identity 44.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 44.1%

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

      3. associate-*l/ 44.1%

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

      4. cos-neg 44.1%

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

      5. associate-*l* 44.1%

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

      6. associate-*r/ 44.1%

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

      7. exp-0 44.1%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 44.1%

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

      9. *-commutative 44.1%

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

      10. neg-sub0 44.1%

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

      11. cos-neg 44.1%

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

   3. Simplified 44.1%

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

   5. Taylor expanded in im around 0 62.1%

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

   6. Taylor expanded in re around 0 34.7%

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

      1. *-commutative 34.7%

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

   8. Simplified 34.7%

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

   if 7.79999999999999986e-4 < im

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 99.9%

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

      3. associate-*l/ 99.9%

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

      4. cos-neg 99.9%

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

      5. associate-*l* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. exp-0 99.9%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 99.9%

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

      9. *-commutative 99.9%

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

      10. neg-sub0 99.9%

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

      11. cos-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in im around 0 84.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 84.4%

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

      2. associate-+r+ 84.4%

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

      3. associate-*r* 84.4%

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

      4. *-commutative 84.4%

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

      5. distribute-rgt-out 84.4%

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

      6. +-commutative 84.4%

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

      7. metadata-eval 84.4%

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

      8. sub-neg 84.4%

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

      9. fma-define 84.4%

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

   7. Simplified 84.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(\cos re, \mathsf{fma}\left(-0.3333333333333333, {im}^{2}, -2\right), \cos re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0003968253968253968, -0.016666666666666666\right) \cdot {im}^{4}\right)\right)\right)} \]

   8. Taylor expanded in im around inf 84.3%

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

      1. associate-*r* 84.3%

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

      2. *-commutative 84.3%

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

   10. Simplified 84.3%

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

   11. Taylor expanded in re around 0 59.7%

       \[\leadsto \color{blue}{-0.0001984126984126984 \cdot {im}^{7}} \]
   12. Step-by-step derivation

       1. *-commutative 59.7%

          \[\leadsto \color{blue}{{im}^{7} \cdot -0.0001984126984126984} \]

   13. Simplified 59.7%

       \[\leadsto \color{blue}{{im}^{7} \cdot -0.0001984126984126984} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 29.8% 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 58.5%

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

   1. /-rgt-identity 58.5%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 58.5%

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

   3. associate-*l/ 58.5%

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

   4. cos-neg 58.5%

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

   5. associate-*l* 58.5%

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

   6. associate-*r/ 58.5%

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

   7. exp-0 58.5%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 58.5%

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

   9. *-commutative 58.5%

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

   10. neg-sub0 58.5%

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

   11. cos-neg 58.5%

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

3. Simplified 58.5%

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

5. Taylor expanded in im around 0 47.5%

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

6. Taylor expanded in re around 0 26.7%

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

   1. *-commutative 26.7%

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

8. Simplified 26.7%

   \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot -2\right)} \]
9. 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 2024107 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (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))))