2-ancestry mixing, zero discriminant

Percentage Accurate: 75.8% → 98.7%

Time: 7.9s

Alternatives: 9

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 75.8% 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: 98.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

   1. div-inv 71.3%

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

   2. cbrt-prod 98.8%

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

   3. associate-/r* 98.8%

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

   4. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

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

   1. div-inv 98.8%

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

   2. cbrt-prod 98.8%

      \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{1}{a}}\right)} \]

5. Applied egg-rr 98.8%

   \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{1}{a}}\right)} \]

6. Final simplification 98.8%

   \[\leadsto \sqrt[3]{g} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{1}{a}}\right) \]

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

   1. div-inv 71.3%

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

   2. cbrt-prod 98.8%

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

   3. associate-/r* 98.8%

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

   4. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

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

4. Final simplification 98.8%

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

Alternative 3: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

   1. clear-num 70.6%

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

   2. cbrt-div 72.8%

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

   3. metadata-eval 72.8%

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

   4. *-un-lft-identity 72.8%

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

   5. times-frac 72.7%

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

   6. metadata-eval 72.7%

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

3. Applied egg-rr 72.7%

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

   1. associate-*r/ 72.8%

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

   2. associate-*l/ 72.8%

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

5. Simplified 72.8%

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

6. Final simplification 72.8%

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

Alternative 4: 7.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\frac{g}{-2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= a -4e-310) (cbrt (/ g -2.0)) (cbrt g)))

C

double code(double g, double a) {
	double tmp;
	if (a <= -4e-310) {
		tmp = cbrt((g / -2.0));
	} else {
		tmp = cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if (a <= -4e-310) {
		tmp = Math.cbrt((g / -2.0));
	} else {
		tmp = Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (a <= -4e-310)
		tmp = cbrt(Float64(g / -2.0));
	else
		tmp = cbrt(g);
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[a, -4e-310], N[Power[N[(g / -2.0), $MachinePrecision], 1/3], $MachinePrecision], N[Power[g, 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.999999999999988e-310

   1. Initial program 61.4%

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

      1. clear-num 60.1%

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

      2. cbrt-div 61.8%

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

      3. metadata-eval 61.8%

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

      4. *-un-lft-identity 61.8%

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

      5. times-frac 61.7%

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

      6. metadata-eval 61.7%

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

   3. Applied egg-rr 61.7%

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

      1. associate-*r/ 61.8%

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

      2. associate-*l/ 61.9%

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

   5. Simplified 61.9%

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

      1. associate-*l/ 61.8%

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

      2. *-commutative 61.8%

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

      3. cbrt-div 98.7%

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

      4. div-inv 98.6%

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

      5. metadata-eval 98.6%

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

      6. div-inv 98.6%

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

      7. add-log-exp 2.6%

         \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\log \left(e^{\frac{a}{0.5}}\right)}} \cdot \frac{1}{\sqrt[3]{g}}} \]

      8. div-inv 2.6%

         \[\leadsto \frac{1}{\sqrt[3]{\log \left(e^{\color{blue}{a \cdot \frac{1}{0.5}}}\right)} \cdot \frac{1}{\sqrt[3]{g}}} \]

      9. metadata-eval 2.6%

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

      10. exp-lft-sqr 2.6%

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

      11. log-prod 2.6%

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

      12. add-log-exp 12.4%

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

      13. add-log-exp 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.7%

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

      2. *-rgt-identity 98.7%

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

   9. Simplified 98.7%

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

       1. clear-num 98.7%

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

       2. cbrt-undiv 61.4%

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

   11. Applied egg-rr 61.4%

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

   13. Simplified 7.4%

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

   if -3.999999999999988e-310 < a

   1. Initial program 81.8%

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

      1. expm1-log1p-u 55.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)\right)} \]

      2. expm1-udef 26.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)} - 1} \]

      3. log1p-udef 26.4%

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

      4. add-exp-log 52.5%

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

      5. *-un-lft-identity 52.5%

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

      6. times-frac 52.5%

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

      7. metadata-eval 52.5%

         \[\leadsto \left(1 + \sqrt[3]{\color{blue}{0.5} \cdot \frac{g}{a}}\right) - 1 \]

   3. Applied egg-rr 52.5%

      \[\leadsto \color{blue}{\left(1 + \sqrt[3]{0.5 \cdot \frac{g}{a}}\right) - 1} \]
   4. Step-by-step derivation

      1. +-commutative 52.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.5 \cdot \frac{g}{a}} + 1\right)} - 1 \]

      2. associate--l+ 81.8%

         \[\leadsto \color{blue}{\sqrt[3]{0.5 \cdot \frac{g}{a}} + \left(1 - 1\right)} \]

      3. metadata-eval 81.8%

         \[\leadsto \sqrt[3]{0.5 \cdot \frac{g}{a}} + \color{blue}{0} \]

      4. +-rgt-identity 81.8%

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

      5. associate-*r/ 81.8%

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

      6. associate-*l/ 81.7%

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

   5. Simplified 81.7%

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

      1. associate-*l/ 81.8%

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

      2. associate-/l* 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. associate-/r/ 81.7%

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

      2. *-commutative 81.7%

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

      3. cbrt-unprod 98.8%

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

      4. /-rgt-identity 98.8%

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

      5. associate-/r/ 98.7%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}} \]

      6. frac-2neg 98.7%

         \[\leadsto \color{blue}{\frac{-\sqrt[3]{g}}{-\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}} \]

      7. div-inv 98.7%

         \[\leadsto \color{blue}{\left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}} \]

      8. add-cube-cbrt 97.7%

         \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\color{blue}{\left(\sqrt[3]{\frac{1}{\sqrt[3]{\frac{0.5}{a}}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}\right) \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}}} \]

      9. add-cube-cbrt 98.7%

         \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\color{blue}{\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}} \]

      10. div-inv 98.7%

          \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\sqrt[3]{\color{blue}{0.5 \cdot \frac{1}{a}}}}} \]

      11. cbrt-unprod 98.8%

          \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\color{blue}{\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{1}{a}}}}} \]

      12. cbrt-unprod 98.7%

          \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\color{blue}{\sqrt[3]{0.5 \cdot \frac{1}{a}}}}} \]

      13. div-inv 98.7%

          \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\sqrt[3]{\color{blue}{\frac{0.5}{a}}}}} \]

      14. clear-num 98.7%

          \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\sqrt[3]{\color{blue}{\frac{1}{\frac{a}{0.5}}}}}} \]

      15. cbrt-div 98.8%

          \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{a}{0.5}}}}}} \]

      16. metadata-eval 98.8%

          \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\frac{\color{blue}{1}}{\sqrt[3]{\frac{a}{0.5}}}}} \]

      17. div-inv 98.8%

          \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\frac{1}{\sqrt[3]{\color{blue}{a \cdot \frac{1}{0.5}}}}}} \]

      18. metadata-eval 98.8%

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

      19. *-commutative 98.8%

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

      20. count-2 98.8%

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

      21. flip-+ 0.0%

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

   9. Applied egg-rr 4.8%

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

   11. Simplified 8.0%

       \[\leadsto \color{blue}{\sqrt[3]{g}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\frac{g}{-2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g}\\ \end{array} \]

Alternative 5: 75.8% 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 71.3%

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

   1. expm1-log1p-u 52.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)\right)} \]

   2. expm1-udef 23.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)} - 1} \]

   3. log1p-udef 23.8%

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

   4. add-exp-log 42.4%

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

   5. *-un-lft-identity 42.4%

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

   6. times-frac 42.4%

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

   7. metadata-eval 42.4%

      \[\leadsto \left(1 + \sqrt[3]{\color{blue}{0.5} \cdot \frac{g}{a}}\right) - 1 \]

3. Applied egg-rr 42.4%

   \[\leadsto \color{blue}{\left(1 + \sqrt[3]{0.5 \cdot \frac{g}{a}}\right) - 1} \]
4. Step-by-step derivation

   1. +-commutative 42.4%

      \[\leadsto \color{blue}{\left(\sqrt[3]{0.5 \cdot \frac{g}{a}} + 1\right)} - 1 \]

   2. associate--l+ 71.3%

      \[\leadsto \color{blue}{\sqrt[3]{0.5 \cdot \frac{g}{a}} + \left(1 - 1\right)} \]

   3. metadata-eval 71.3%

      \[\leadsto \sqrt[3]{0.5 \cdot \frac{g}{a}} + \color{blue}{0} \]

   4. +-rgt-identity 71.3%

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

   5. associate-*r/ 71.3%

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

   6. associate-*l/ 71.3%

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

5. Simplified 71.3%

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

6. Final simplification 71.3%

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

Alternative 6: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.3%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]

2. Final simplification 71.3%

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

Alternative 7: 17.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

   1. clear-num 70.6%

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

   2. cbrt-div 72.8%

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

   3. metadata-eval 72.8%

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

   4. *-un-lft-identity 72.8%

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

   5. times-frac 72.7%

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

   6. metadata-eval 72.7%

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

3. Applied egg-rr 72.7%

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

   1. associate-*r/ 72.8%

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

   2. associate-*l/ 72.8%

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

5. Simplified 72.8%

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

   1. associate-*l/ 72.8%

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

   2. *-commutative 72.8%

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

   3. cbrt-div 98.7%

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

   4. div-inv 98.7%

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

   5. metadata-eval 98.7%

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

   6. div-inv 98.7%

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

   7. add-log-exp 4.0%

      \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\log \left(e^{\frac{a}{0.5}}\right)}} \cdot \frac{1}{\sqrt[3]{g}}} \]

   8. div-inv 4.0%

      \[\leadsto \frac{1}{\sqrt[3]{\log \left(e^{\color{blue}{a \cdot \frac{1}{0.5}}}\right)} \cdot \frac{1}{\sqrt[3]{g}}} \]

   9. metadata-eval 4.0%

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

   10. exp-lft-sqr 3.9%

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

   11. log-prod 3.9%

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

   12. add-log-exp 13.5%

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

   13. add-log-exp 98.7%

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

7. Applied egg-rr 98.7%

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

   1. associate-*r/ 98.7%

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

   2. *-rgt-identity 98.7%

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

9. Simplified 98.7%

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

    1. clear-num 98.8%

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

    2. count-2 98.8%

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

    3. cbrt-prod 98.3%

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

    4. *-commutative 98.3%

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

    5. associate-/l/ 98.3%

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

    6. cbrt-undiv 98.7%

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

    7. cbrt-undiv 71.3%

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

    8. div-inv 71.3%

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

    9. metadata-eval 71.3%

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

11. Applied egg-rr 71.3%

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

13. Simplified 16.3%

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

14. Final simplification 16.3%

    \[\leadsto \sqrt[3]{\frac{g}{a}} \]

Alternative 8: 4.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{g} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt g))

C

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

Java

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

Julia

function code(g, a)
	return cbrt(g)
end

Wolfram

code[g_, a_] := N[Power[g, 1/3], $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. expm1-log1p-u 52.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)\right)} \]

   2. expm1-udef 23.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{2 \cdot a}}\right)} - 1} \]

   3. log1p-udef 23.8%

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

   4. add-exp-log 42.4%

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

   5. *-un-lft-identity 42.4%

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

   6. times-frac 42.4%

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

   7. metadata-eval 42.4%

      \[\leadsto \left(1 + \sqrt[3]{\color{blue}{0.5} \cdot \frac{g}{a}}\right) - 1 \]

3. Applied egg-rr 42.4%

   \[\leadsto \color{blue}{\left(1 + \sqrt[3]{0.5 \cdot \frac{g}{a}}\right) - 1} \]
4. Step-by-step derivation

   1. +-commutative 42.4%

      \[\leadsto \color{blue}{\left(\sqrt[3]{0.5 \cdot \frac{g}{a}} + 1\right)} - 1 \]

   2. associate--l+ 71.3%

      \[\leadsto \color{blue}{\sqrt[3]{0.5 \cdot \frac{g}{a}} + \left(1 - 1\right)} \]

   3. metadata-eval 71.3%

      \[\leadsto \sqrt[3]{0.5 \cdot \frac{g}{a}} + \color{blue}{0} \]

   4. +-rgt-identity 71.3%

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

   5. associate-*r/ 71.3%

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

   6. associate-*l/ 71.3%

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

5. Simplified 71.3%

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

   1. associate-*l/ 71.3%

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

   2. associate-/l* 70.6%

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

7. Applied egg-rr 70.6%

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

   1. associate-/r/ 71.3%

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

   2. *-commutative 71.3%

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

   3. cbrt-unprod 98.8%

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

   4. /-rgt-identity 98.8%

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

   5. associate-/r/ 98.7%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}} \]

   6. frac-2neg 98.7%

      \[\leadsto \color{blue}{\frac{-\sqrt[3]{g}}{-\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}} \]

   7. div-inv 98.8%

      \[\leadsto \color{blue}{\left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}} \]

   8. add-cube-cbrt 97.8%

      \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\color{blue}{\left(\sqrt[3]{\frac{1}{\sqrt[3]{\frac{0.5}{a}}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}\right) \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}}} \]

   9. add-cube-cbrt 98.8%

      \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\color{blue}{\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}} \]

   10. div-inv 98.8%

       \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\sqrt[3]{\color{blue}{0.5 \cdot \frac{1}{a}}}}} \]

   11. cbrt-unprod 98.8%

       \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\color{blue}{\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{1}{a}}}}} \]

   12. cbrt-unprod 98.8%

       \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\color{blue}{\sqrt[3]{0.5 \cdot \frac{1}{a}}}}} \]

   13. div-inv 98.8%

       \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\sqrt[3]{\color{blue}{\frac{0.5}{a}}}}} \]

   14. clear-num 98.8%

       \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\sqrt[3]{\color{blue}{\frac{1}{\frac{a}{0.5}}}}}} \]

   15. cbrt-div 98.8%

       \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{a}{0.5}}}}}} \]

   16. metadata-eval 98.8%

       \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\frac{\color{blue}{1}}{\sqrt[3]{\frac{a}{0.5}}}}} \]

   17. div-inv 98.8%

       \[\leadsto \left(-\sqrt[3]{g}\right) \cdot \frac{1}{-\frac{1}{\frac{1}{\sqrt[3]{\color{blue}{a \cdot \frac{1}{0.5}}}}}} \]

   18. metadata-eval 98.8%

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

   19. *-commutative 98.8%

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

   20. count-2 98.8%

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

   21. flip-+ 0.0%

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

9. Applied egg-rr 4.6%

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

11. Simplified 4.7%

    \[\leadsto \color{blue}{\sqrt[3]{g}} \]

12. Final simplification 4.7%

    \[\leadsto \sqrt[3]{g} \]

Alternative 9: 4.5% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (g a) :precision binary64 -2.0)

C

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

Fortran

real(8) function code(g, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: a
    code = -2.0d0
end function

Java

public static double code(double g, double a) {
	return -2.0;
}

Python

def code(g, a):
	return -2.0

Julia

function code(g, a)
	return -2.0
end

MATLAB

function tmp = code(g, a)
	tmp = -2.0;
end

Wolfram

code[g_, a_] := -2.0

Derivation

1. Initial program 71.3%

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

   1. div-inv 71.3%

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

   2. cbrt-prod 98.8%

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

   3. associate-/r* 98.8%

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

   4. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

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

   1. div-inv 98.8%

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

   2. cbrt-prod 98.8%

      \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{1}{a}}\right)} \]

5. Applied egg-rr 98.8%

   \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{1}{a}}\right)} \]
6. Step-by-step derivation

   1. cbrt-unprod 98.8%

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

   2. div-inv 98.8%

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

   3. /-rgt-identity 98.8%

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

   4. associate-/r/ 98.7%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}} \]

   5. clear-num 98.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{\sqrt[3]{\frac{0.5}{a}}}}{\sqrt[3]{g}}}} \]

7. Applied egg-rr 6.2%

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

8. Taylor expanded in a around 0 3.0%

   \[\leadsto \color{blue}{0.5 \cdot \left({\left(1 \cdot g\right)}^{0.3333333333333333} \cdot \frac{1}{a}\right)} \]

10. Simplified 4.3%

    \[\leadsto \color{blue}{-2} \]

11. Final simplification 4.3%

    \[\leadsto -2 \]

Reproduce

herbie shell --seed 2023175 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))