2-ancestry mixing, zero discriminant

Percentage Accurate: 76.9% → 98.6%

Time: 8.0s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{2 \cdot a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ g (* 2.0 a))))

C

double code(double g, double a) {
	return cbrt((g / (2.0 * a)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g / (2.0 * a)));
}

Julia

function code(g, a)
	return cbrt(Float64(g / Float64(2.0 * a)))
end

Wolfram

code[g_, a_] := N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Initial Program: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{2 \cdot a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ g (* 2.0 a))))

C

double code(double g, double a) {
	return cbrt((g / (2.0 * a)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g / (2.0 * a)));
}

Julia

function code(g, a)
	return cbrt(Float64(g / Float64(2.0 * a)))
end

Wolfram

code[g_, a_] := N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Alternative 1: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\frac{-2}{g}}} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (* (cbrt (/ -1.0 a)) (/ 1.0 (cbrt (/ -2.0 g)))))

C

double code(double g, double a) {
	return cbrt((-1.0 / a)) * (1.0 / cbrt((-2.0 / g)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((-1.0 / a)) * (1.0 / Math.cbrt((-2.0 / g)));
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(-1.0 / a)) * Float64(1.0 / cbrt(Float64(-2.0 / g))))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[(1.0 / N[Power[N[(-2.0 / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{2}}{a}}} \]

   2. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{g}{2}}}}{\sqrt[3]{a}} \]

   5. div-inv N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}}}{\sqrt[3]{a}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}}}{\sqrt[3]{a}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\frac{1}{2}}}}{\sqrt[3]{a}} \]

   8. cbrt-lowering-cbrt.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g \cdot 0.5}}{\color{blue}{\sqrt[3]{a}}} \]

4. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot 0.5}}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g \cdot \frac{1}{2}}}}} \]

   2. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{a}} \cdot \sqrt[3]{g \cdot \frac{1}{2}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{1}}}{\sqrt[3]{a}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} \]

   4. cbrt-div N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a}}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{g \cdot \color{blue}{\frac{1}{2}}} \]

   6. div-inv N/A

      \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\color{blue}{\frac{g}{2}}} \]

   7. clear-num N/A

      \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{2}{g}}}} \]

   8. cbrt-prod N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a} \cdot \frac{1}{\frac{2}{g}}}} \]

   9. div-inv N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{a}}{\frac{2}{g}}}} \]

   10. frac-2neg N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{a}\right)}{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   11. cbrt-div N/A

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   12. pow1/3 N/A

       \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)}^{\frac{1}{3}}}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   13. div-inv N/A

       \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)}^{\frac{1}{3}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)}^{\frac{1}{3}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   15. pow1/3 N/A

       \[\leadsto \color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   16. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   17. distribute-neg-frac N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{a}}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   18. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{\color{blue}{-1}}{a}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   19. /-lowering-/.f64 N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{a}}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \color{blue}{\frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   21. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   22. distribute-neg-frac N/A

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{g}}}} \]

   23. /-lowering-/.f64 N/A

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{g}}}} \]

   24. metadata-eval 98.8

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\frac{\color{blue}{-2}}{g}}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\frac{-2}{g}}}} \]
7. Add Preprocessing

Alternative 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{\frac{1}{\frac{2}{g}}}}{\sqrt[3]{a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ (cbrt (/ 1.0 (/ 2.0 g))) (cbrt a)))

C

double code(double g, double a) {
	return cbrt((1.0 / (2.0 / g))) / cbrt(a);
}

Java

public static double code(double g, double a) {
	return Math.cbrt((1.0 / (2.0 / g))) / Math.cbrt(a);
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(1.0 / Float64(2.0 / g))) / cbrt(a))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(1.0 / N[(2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{2}}{a}}} \]

   2. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{g}{2}}}}{\sqrt[3]{a}} \]

   5. div-inv N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}}}{\sqrt[3]{a}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}}}{\sqrt[3]{a}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\frac{1}{2}}}}{\sqrt[3]{a}} \]

   8. cbrt-lowering-cbrt.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g \cdot 0.5}}{\color{blue}{\sqrt[3]{a}}} \]

4. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot 0.5}}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\frac{1}{2}}}}{\sqrt[3]{a}} \]

   2. div-inv N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g}{2}}}}{\sqrt[3]{a}} \]

   3. clear-num N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{1}{\frac{2}{g}}}}}{\sqrt[3]{a}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{1}{\frac{2}{g}}}}}{\sqrt[3]{a}} \]

   5. /-lowering-/.f64 98.8

      \[\leadsto \frac{\sqrt[3]{\frac{1}{\color{blue}{\frac{2}{g}}}}}{\sqrt[3]{a}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{1}{\frac{2}{g}}}}}{\sqrt[3]{a}} \]
7. Add Preprocessing

Alternative 3: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{\frac{-1}{a}}}{\sqrt[3]{\frac{-2}{g}}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ (cbrt (/ -1.0 a)) (cbrt (/ -2.0 g))))

C

double code(double g, double a) {
	return cbrt((-1.0 / a)) / cbrt((-2.0 / g));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((-1.0 / a)) / Math.cbrt((-2.0 / g));
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(-1.0 / a)) / cbrt(Float64(-2.0 / g)))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(-1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(-2.0 / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{2}}{a}}} \]

   2. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{g}{2}}}}{\sqrt[3]{a}} \]

   5. div-inv N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}}}{\sqrt[3]{a}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}}}{\sqrt[3]{a}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\frac{1}{2}}}}{\sqrt[3]{a}} \]

   8. cbrt-lowering-cbrt.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g \cdot 0.5}}{\color{blue}{\sqrt[3]{a}}} \]

4. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot 0.5}}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g \cdot \frac{1}{2}}}}} \]

   2. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{a}} \cdot \sqrt[3]{g \cdot \frac{1}{2}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{1}}}{\sqrt[3]{a}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} \]

   4. cbrt-div N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a}}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{g \cdot \color{blue}{\frac{1}{2}}} \]

   6. div-inv N/A

      \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\color{blue}{\frac{g}{2}}} \]

   7. clear-num N/A

      \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{2}{g}}}} \]

   8. cbrt-prod N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a} \cdot \frac{1}{\frac{2}{g}}}} \]

   9. div-inv N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{a}}{\frac{2}{g}}}} \]

   10. frac-2neg N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{a}\right)}{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   11. cbrt-div N/A

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   12. pow1/3 N/A

       \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)}^{\frac{1}{3}}}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \color{blue}{\frac{{\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)}^{\frac{1}{3}}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   14. pow1/3 N/A

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)}}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   15. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)}}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   16. distribute-neg-frac N/A

       \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{a}}}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   17. metadata-eval N/A

       \[\leadsto \frac{\sqrt[3]{\frac{\color{blue}{-1}}{a}}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   18. /-lowering-/.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{-1}{a}}}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   19. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\frac{-1}{a}}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   20. distribute-neg-frac N/A

       \[\leadsto \frac{\sqrt[3]{\frac{-1}{a}}}{\sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{g}}}} \]

   21. /-lowering-/.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\frac{-1}{a}}}{\sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{g}}}} \]

   22. metadata-eval 98.8

       \[\leadsto \frac{\sqrt[3]{\frac{-1}{a}}}{\sqrt[3]{\frac{\color{blue}{-2}}{g}}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{-1}{a}}}{\sqrt[3]{\frac{-2}{g}}}} \]
7. Add Preprocessing

Alternative 4: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 5 \cdot 10^{-294}:\\ \;\;\;\;\sqrt[3]{0.5 \cdot \frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) 5e-294)
   (cbrt (* 0.5 (/ g a)))
   (* (cbrt (* g 0.5)) (pow a -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 5e-294) {
		tmp = cbrt((0.5 * (g / a)));
	} else {
		tmp = cbrt((g * 0.5)) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 5e-294) {
		tmp = Math.cbrt((0.5 * (g / a)));
	} else {
		tmp = Math.cbrt((g * 0.5)) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= 5e-294)
		tmp = cbrt(Float64(0.5 * Float64(g / a)));
	else
		tmp = Float64(cbrt(Float64(g * 0.5)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], 5e-294], N[Power[N[(0.5 * N[(g / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < 5.0000000000000003e-294

   1. Initial program 78.2%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{a}}{2}}} \]

      2. div-inv N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a} \cdot \frac{1}{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a} \cdot \frac{1}{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a}} \cdot \frac{1}{2}} \]

      5. metadata-eval 78.2

         \[\leadsto \sqrt[3]{\frac{g}{a} \cdot \color{blue}{0.5}} \]

   4. Applied egg-rr 78.2%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a} \cdot 0.5}} \]

   if 5.0000000000000003e-294 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 78.1%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{2}}{a}}} \]

      2. div-inv N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{2} \cdot \frac{1}{a}}} \]

      3. cbrt-prod N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2}} \cdot \sqrt[3]{\frac{1}{a}}} \]

      4. pow1/3 N/A

         \[\leadsto \sqrt[3]{\frac{g}{2}} \cdot \color{blue}{{\left(\frac{1}{a}\right)}^{\frac{1}{3}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{a}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\frac{g}{2}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{a}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\frac{g}{2}}} \]

      7. inv-pow N/A

         \[\leadsto {\color{blue}{\left({a}^{-1}\right)}}^{\frac{1}{3}} \cdot \sqrt[3]{\frac{g}{2}} \]

      8. pow-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(-1 \cdot \frac{1}{3}\right)}} \cdot \sqrt[3]{\frac{g}{2}} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(-1 \cdot \frac{1}{3}\right)}} \cdot \sqrt[3]{\frac{g}{2}} \]

      10. metadata-eval N/A

          \[\leadsto {a}^{\color{blue}{\frac{-1}{3}}} \cdot \sqrt[3]{\frac{g}{2}} \]

      11. cbrt-lowering-cbrt.f64 N/A

          \[\leadsto {a}^{\frac{-1}{3}} \cdot \color{blue}{\sqrt[3]{\frac{g}{2}}} \]

      12. div-inv N/A

          \[\leadsto {a}^{\frac{-1}{3}} \cdot \sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto {a}^{\frac{-1}{3}} \cdot \sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}} \]

      14. metadata-eval 92.4

          \[\leadsto {a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot \color{blue}{0.5}} \]

   4. Applied egg-rr 92.4%

      \[\leadsto \color{blue}{{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 5 \cdot 10^{-294}:\\ \;\;\;\;\sqrt[3]{0.5 \cdot \frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{g \cdot 0.5}}{\sqrt[3]{a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ (cbrt (* g 0.5)) (cbrt a)))

C

double code(double g, double a) {
	return cbrt((g * 0.5)) / cbrt(a);
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g * 0.5)) / Math.cbrt(a);
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(g * 0.5)) / cbrt(a))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{2}}{a}}} \]

   2. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{g}{2}}}}{\sqrt[3]{a}} \]

   5. div-inv N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}}}{\sqrt[3]{a}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}}}{\sqrt[3]{a}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\frac{1}{2}}}}{\sqrt[3]{a}} \]

   8. cbrt-lowering-cbrt.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g \cdot 0.5}}{\color{blue}{\sqrt[3]{a}}} \]

4. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot 0.5}}{\sqrt[3]{a}}} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (* (cbrt (/ 0.5 a)) (cbrt g)))

C

double code(double g, double a) {
	return cbrt((0.5 / a)) * cbrt(g);
}

Java

public static double code(double g, double a) {
	return Math.cbrt((0.5 / a)) * Math.cbrt(g);
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(0.5 / a)) * cbrt(g))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{2}}{a}}} \]

   2. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{g}{2}}}}{\sqrt[3]{a}} \]

   5. div-inv N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}}}{\sqrt[3]{a}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}}}{\sqrt[3]{a}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\frac{1}{2}}}}{\sqrt[3]{a}} \]

   8. cbrt-lowering-cbrt.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g \cdot 0.5}}{\color{blue}{\sqrt[3]{a}}} \]

4. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot 0.5}}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g \cdot \frac{1}{2}}}}} \]

   2. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{a}} \cdot \sqrt[3]{g \cdot \frac{1}{2}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{1}}}{\sqrt[3]{a}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} \]

   4. cbrt-div N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a}}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{g \cdot \color{blue}{\frac{1}{2}}} \]

   6. div-inv N/A

      \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\color{blue}{\frac{g}{2}}} \]

   7. clear-num N/A

      \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{2}{g}}}} \]

   8. cbrt-prod N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a} \cdot \frac{1}{\frac{2}{g}}}} \]

   9. div-inv N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{a}}{\frac{2}{g}}}} \]

   10. frac-2neg N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{a}\right)}{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   11. cbrt-div N/A

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   12. pow1/3 N/A

       \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)}^{\frac{1}{3}}}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   13. div-inv N/A

       \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)}^{\frac{1}{3}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)}^{\frac{1}{3}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   15. pow1/3 N/A

       \[\leadsto \color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   16. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   17. distribute-neg-frac N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{a}}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   18. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{\color{blue}{-1}}{a}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   19. /-lowering-/.f64 N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{a}}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \color{blue}{\frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   21. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   22. distribute-neg-frac N/A

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{g}}}} \]

   23. /-lowering-/.f64 N/A

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{g}}}} \]

   24. metadata-eval 98.8

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\frac{\color{blue}{-2}}{g}}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\frac{-2}{g}}}} \]
7. Step-by-step derivation

   1. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{-1}{a}}}{\sqrt[3]{\frac{-2}{g}}}} \]

   2. cbrt-undiv N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{-1}{a}}{\frac{-2}{g}}}} \]

   3. clear-num N/A

      \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{\frac{a}{-1}}}}{\frac{-2}{g}}} \]

   4. associate-/r/ N/A

      \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{a} \cdot -1}}{\frac{-2}{g}}} \]

   5. associate-/l* N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{a} \cdot \frac{-1}{\frac{-2}{g}}}} \]

   6. inv-pow N/A

      \[\leadsto \sqrt[3]{\color{blue}{{a}^{-1}} \cdot \frac{-1}{\frac{-2}{g}}} \]

   7. metadata-eval N/A

      \[\leadsto \sqrt[3]{{a}^{\color{blue}{\left(2 - 3\right)}} \cdot \frac{-1}{\frac{-2}{g}}} \]

   8. pow-div N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{{a}^{2}}{{a}^{3}}} \cdot \frac{-1}{\frac{-2}{g}}} \]

   9. pow2 N/A

      \[\leadsto \sqrt[3]{\frac{\color{blue}{a \cdot a}}{{a}^{3}} \cdot \frac{-1}{\frac{-2}{g}}} \]

   10. cube-unmult N/A

       \[\leadsto \sqrt[3]{\frac{a \cdot a}{\color{blue}{a \cdot \left(a \cdot a\right)}} \cdot \frac{-1}{\frac{-2}{g}}} \]

   11. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{a \cdot a}{a \cdot \left(a \cdot a\right)} \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{\frac{-2}{g}}} \]

   12. distribute-neg-frac N/A

       \[\leadsto \sqrt[3]{\frac{a \cdot a}{a \cdot \left(a \cdot a\right)} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\frac{-2}{g}}\right)\right)}} \]

   13. clear-num N/A

       \[\leadsto \sqrt[3]{\frac{a \cdot a}{a \cdot \left(a \cdot a\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{g}{-2}}\right)\right)} \]

   14. distribute-neg-frac2 N/A

       \[\leadsto \sqrt[3]{\frac{a \cdot a}{a \cdot \left(a \cdot a\right)} \cdot \color{blue}{\frac{g}{\mathsf{neg}\left(-2\right)}}} \]

   15. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{a \cdot a}{a \cdot \left(a \cdot a\right)} \cdot \frac{g}{\color{blue}{2}}} \]

   16. times-frac N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\left(a \cdot a\right) \cdot g}{\left(a \cdot \left(a \cdot a\right)\right) \cdot 2}}} \]

   17. associate-*r/ N/A

       \[\leadsto \sqrt[3]{\color{blue}{\left(a \cdot a\right) \cdot \frac{g}{\left(a \cdot \left(a \cdot a\right)\right) \cdot 2}}} \]

   18. clear-num N/A

       \[\leadsto \sqrt[3]{\left(a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot 2}{g}}}} \]

   19. un-div-inv N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{a \cdot a}{\frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot 2}{g}}}} \]

   20. associate-/r/ N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{a \cdot a}{\left(a \cdot \left(a \cdot a\right)\right) \cdot 2} \cdot g}} \]

8. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g}} \]
9. Add Preprocessing

Alternative 7: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ 1.0 (cbrt (/ a (* g 0.5)))))

C

double code(double g, double a) {
	return 1.0 / cbrt((a / (g * 0.5)));
}

Java

public static double code(double g, double a) {
	return 1.0 / Math.cbrt((a / (g * 0.5)));
}

Julia

function code(g, a)
	return Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))))
end

Wolfram

code[g_, a_] := N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{2}}{a}}} \]

   2. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{g}{2}}}}{\sqrt[3]{a}} \]

   5. div-inv N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}}}{\sqrt[3]{a}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \frac{1}{2}}}}{\sqrt[3]{a}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\frac{1}{2}}}}{\sqrt[3]{a}} \]

   8. cbrt-lowering-cbrt.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g \cdot 0.5}}{\color{blue}{\sqrt[3]{a}}} \]

4. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot 0.5}}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g \cdot \frac{1}{2}}}}} \]

   2. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{a}} \cdot \sqrt[3]{g \cdot \frac{1}{2}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{1}}}{\sqrt[3]{a}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} \]

   4. cbrt-div N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a}}} \cdot \sqrt[3]{g \cdot \frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{g \cdot \color{blue}{\frac{1}{2}}} \]

   6. div-inv N/A

      \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\color{blue}{\frac{g}{2}}} \]

   7. clear-num N/A

      \[\leadsto \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{2}{g}}}} \]

   8. cbrt-prod N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a} \cdot \frac{1}{\frac{2}{g}}}} \]

   9. div-inv N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{a}}{\frac{2}{g}}}} \]

   10. frac-2neg N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{a}\right)}{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   11. cbrt-div N/A

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   12. pow1/3 N/A

       \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)}^{\frac{1}{3}}}}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   13. div-inv N/A

       \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)}^{\frac{1}{3}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(\frac{1}{a}\right)\right)}^{\frac{1}{3}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   15. pow1/3 N/A

       \[\leadsto \color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   16. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{1}{a}\right)}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   17. distribute-neg-frac N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{a}}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   18. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{\color{blue}{-1}}{a}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   19. /-lowering-/.f64 N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{a}}} \cdot \frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}} \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \color{blue}{\frac{1}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   21. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(\frac{2}{g}\right)}}} \]

   22. distribute-neg-frac N/A

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{g}}}} \]

   23. /-lowering-/.f64 N/A

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{g}}}} \]

   24. metadata-eval 98.8

       \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\frac{\color{blue}{-2}}{g}}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \frac{1}{\sqrt[3]{\frac{-2}{g}}}} \]

8. Applied egg-rr 78.8%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}} \]
9. Add Preprocessing

Alternative 8: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{0.5 \cdot \frac{g}{a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (* 0.5 (/ g a))))

C

double code(double g, double a) {
	return cbrt((0.5 * (g / a)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((0.5 * (g / a)));
}

Julia

function code(g, a)
	return cbrt(Float64(0.5 * Float64(g / a)))
end

Wolfram

code[g_, a_] := N[Power[N[(0.5 * N[(g / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{a}}{2}}} \]

   2. div-inv N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a} \cdot \frac{1}{2}}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a} \cdot \frac{1}{2}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a}} \cdot \frac{1}{2}} \]

   5. metadata-eval 78.5

      \[\leadsto \sqrt[3]{\frac{g}{a} \cdot \color{blue}{0.5}} \]

4. Applied egg-rr 78.5%

   \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a} \cdot 0.5}} \]

5. Final simplification 78.5%

   \[\leadsto \sqrt[3]{0.5 \cdot \frac{g}{a}} \]
6. Add Preprocessing

Alternative 9: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{g \cdot \frac{0.5}{a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (* g (/ 0.5 a))))

C

double code(double g, double a) {
	return cbrt((g * (0.5 / a)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((g * (0.5 / a)));
}

Julia

function code(g, a)
	return cbrt(Float64(g * Float64(0.5 / a)))
end

Wolfram

code[g_, a_] := N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{2 \cdot a}{g}}}} \]

   2. associate-/r/ N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{2 \cdot a} \cdot g}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{2 \cdot a} \cdot g}} \]

   4. associate-/r* N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot g} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot g} \]

   6. metadata-eval 78.5

      \[\leadsto \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot g} \]

4. Applied egg-rr 78.5%

   \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{a} \cdot g}} \]

5. Final simplification 78.5%

   \[\leadsto \sqrt[3]{g \cdot \frac{0.5}{a}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))