2-ancestry mixing, zero discriminant

Percentage Accurate: 76.3% → 98.7%

Time: 7.6s

Alternatives: 5

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: 76.3% 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, 0.5× 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 80.7%

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

   1. clear-num N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{2 \cdot a}{g}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{1}{2 \cdot a} \cdot g\right)\right) \]

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

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), g\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), g\right)\right) \]

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

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), g\right)\right) \]

   6. metadata-eval 80.7%

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), g\right)\right) \]

4. Applied egg-rr 80.7%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \frac{\frac{1}{2}}{a}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \frac{1}{\frac{a}{\frac{1}{2}}}\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{\frac{a}{\frac{1}{2}}}\right)\right) \]

   4. frac-2neg N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(g\right)}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{0 - g}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   6. flip3-- N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{{0}^{3} - {g}^{3}}{0 \cdot 0 + \left(g \cdot g + 0 \cdot g\right)}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - {g}^{3}}{0 \cdot 0 + \left(g \cdot g + 0 \cdot g\right)}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   8. mul0-lft N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 \cdot g - {g}^{3}}{0 \cdot 0 + \left(g \cdot g + 0 \cdot g\right)}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   9. cube-unmult N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 \cdot g - g \cdot \left(g \cdot g\right)}{0 \cdot 0 + \left(g \cdot g + 0 \cdot g\right)}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 \cdot g - g \cdot \left(g \cdot g\right)}{0 + \left(g \cdot g + 0 \cdot g\right)}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   11. mul0-lft N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 \cdot g - g \cdot \left(g \cdot g\right)}{0 + \left(g \cdot g + 0\right)}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   12. +-rgt-identity N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 \cdot g - g \cdot \left(g \cdot g\right)}{0 + g \cdot g}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   13. +-lft-identity N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 \cdot g - g \cdot \left(g \cdot g\right)}{g \cdot g}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   14. frac-sub N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0}{g} - \frac{g \cdot g}{g}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   15. div-sub N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - g \cdot g}{g}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   16. clear-num N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - g \cdot g}{g}}{\mathsf{neg}\left(\frac{1}{\frac{\frac{1}{2}}{a}}\right)}\right)\right) \]

   17. associate-/r/ N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - g \cdot g}{g}}{\mathsf{neg}\left(\frac{1}{\frac{1}{2}} \cdot a\right)}\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - g \cdot g}{g}}{\mathsf{neg}\left(2 \cdot a\right)}\right)\right) \]

   19. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - g \cdot g}{g}}{\left(\mathsf{neg}\left(2\right)\right) \cdot a}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - g \cdot g}{g}}{-2 \cdot a}\right)\right) \]

   21. un-div-inv N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{0 - g \cdot g}{g} \cdot \frac{1}{-2 \cdot a}\right)\right) \]

   22. associate-/l/ N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{0 - g \cdot g}{g} \cdot \frac{\frac{1}{a}}{-2}\right)\right) \]

   23. times-frac N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\left(0 - g \cdot g\right) \cdot \frac{1}{a}}{g \cdot -2}\right)\right) \]

   24. *-commutative N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\left(0 - g \cdot g\right) \cdot \frac{1}{a}}{-2 \cdot g}\right)\right) \]

6. Applied egg-rr 57.0%

   \[\leadsto \sqrt[3]{\color{blue}{\left(0 - g\right) \cdot \left(g \cdot \frac{\frac{-0.5}{g}}{a}\right)}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r/ N/A

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

   4. associate-*l/ N/A

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

   5. cbrt-div N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{\left(g \cdot \frac{\frac{-1}{2}}{g}\right) \cdot \left(\mathsf{neg}\left(g\right)\right)}\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right) \]

8. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{\frac{g}{\frac{g}{-0.5}}}{\frac{-1}{g}}}}{\sqrt[3]{a}}} \]
9. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{g \cdot \frac{1}{\frac{-1}{2}}}}{\frac{-1}{g}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{g \cdot -2}}{\frac{-1}{g}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{g \cdot \left(\mathsf{neg}\left(2\right)\right)}}{\frac{-1}{g}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{\frac{g}{g}}{\mathsf{neg}\left(2\right)}}{\frac{-1}{g}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{1}{\mathsf{neg}\left(2\right)}}{\frac{-1}{g}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   6. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{-1}{g} \cdot \left(\mathsf{neg}\left(2\right)\right)}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   7. frac-2neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(g\right)} \cdot \left(\mathsf{neg}\left(2\right)\right)}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{1}{\mathsf{neg}\left(g\right)} \cdot \left(\mathsf{neg}\left(2\right)\right)}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   9. associate-/r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{1}{\frac{\mathsf{neg}\left(g\right)}{\mathsf{neg}\left(2\right)}}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   10. frac-2neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{1}{\frac{g}{2}}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{g}{2}}\right)\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   12. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2}{g}\right)\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   13. /-lowering-/.f64 98.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(2, g\right)\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

10. Applied egg-rr 98.7%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. clear-num N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{2 \cdot a}{g}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{1}{2 \cdot a} \cdot g\right)\right) \]

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

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), g\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), g\right)\right) \]

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

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), g\right)\right) \]

   6. metadata-eval 80.7%

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), g\right)\right) \]

4. Applied egg-rr 80.7%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \frac{\frac{1}{2}}{a}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \frac{1}{\frac{a}{\frac{1}{2}}}\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{\frac{a}{\frac{1}{2}}}\right)\right) \]

   4. frac-2neg N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(g\right)}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{0 - g}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   6. flip3-- N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{{0}^{3} - {g}^{3}}{0 \cdot 0 + \left(g \cdot g + 0 \cdot g\right)}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - {g}^{3}}{0 \cdot 0 + \left(g \cdot g + 0 \cdot g\right)}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   8. mul0-lft N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 \cdot g - {g}^{3}}{0 \cdot 0 + \left(g \cdot g + 0 \cdot g\right)}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   9. cube-unmult N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 \cdot g - g \cdot \left(g \cdot g\right)}{0 \cdot 0 + \left(g \cdot g + 0 \cdot g\right)}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 \cdot g - g \cdot \left(g \cdot g\right)}{0 + \left(g \cdot g + 0 \cdot g\right)}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   11. mul0-lft N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 \cdot g - g \cdot \left(g \cdot g\right)}{0 + \left(g \cdot g + 0\right)}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   12. +-rgt-identity N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 \cdot g - g \cdot \left(g \cdot g\right)}{0 + g \cdot g}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   13. +-lft-identity N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 \cdot g - g \cdot \left(g \cdot g\right)}{g \cdot g}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   14. frac-sub N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0}{g} - \frac{g \cdot g}{g}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   15. div-sub N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - g \cdot g}{g}}{\mathsf{neg}\left(\frac{a}{\frac{1}{2}}\right)}\right)\right) \]

   16. clear-num N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - g \cdot g}{g}}{\mathsf{neg}\left(\frac{1}{\frac{\frac{1}{2}}{a}}\right)}\right)\right) \]

   17. associate-/r/ N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - g \cdot g}{g}}{\mathsf{neg}\left(\frac{1}{\frac{1}{2}} \cdot a\right)}\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - g \cdot g}{g}}{\mathsf{neg}\left(2 \cdot a\right)}\right)\right) \]

   19. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - g \cdot g}{g}}{\left(\mathsf{neg}\left(2\right)\right) \cdot a}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\frac{0 - g \cdot g}{g}}{-2 \cdot a}\right)\right) \]

   21. un-div-inv N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{0 - g \cdot g}{g} \cdot \frac{1}{-2 \cdot a}\right)\right) \]

   22. associate-/l/ N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{0 - g \cdot g}{g} \cdot \frac{\frac{1}{a}}{-2}\right)\right) \]

   23. times-frac N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\left(0 - g \cdot g\right) \cdot \frac{1}{a}}{g \cdot -2}\right)\right) \]

   24. *-commutative N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{\left(0 - g \cdot g\right) \cdot \frac{1}{a}}{-2 \cdot g}\right)\right) \]

6. Applied egg-rr 57.0%

   \[\leadsto \sqrt[3]{\color{blue}{\left(0 - g\right) \cdot \left(g \cdot \frac{\frac{-0.5}{g}}{a}\right)}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r/ N/A

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

   4. associate-*l/ N/A

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

   5. cbrt-div N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{\left(g \cdot \frac{\frac{-1}{2}}{g}\right) \cdot \left(\mathsf{neg}\left(g\right)\right)}\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right) \]

8. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{\frac{g}{\frac{g}{-0.5}}}{\frac{-1}{g}}}}{\sqrt[3]{a}}} \]
9. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{g \cdot \frac{1}{\frac{-1}{2}}}}{\frac{-1}{g}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{g \cdot -2}}{\frac{-1}{g}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{g \cdot \left(\mathsf{neg}\left(2\right)\right)}}{\frac{-1}{g}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{\frac{g}{g}}{\mathsf{neg}\left(2\right)}}{\frac{-1}{g}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{1}{\mathsf{neg}\left(2\right)}}{\frac{-1}{g}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{-1}{g}}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   7. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{1}{\frac{-1}{g}}}{\mathsf{neg}\left(2\right)}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{g}{-1}}{\mathsf{neg}\left(2\right)}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   9. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{\left(\mathsf{neg}\left(2\right)\right) \cdot -1}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{-2 \cdot -1}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{2}\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

   12. /-lowering-/.f64 98.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, 2\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

10. Applied egg-rr 98.7%

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

Alternative 3: 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 80.7%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. cbrt-prod N/A

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

   4. cbrt-div N/A

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

   5. metadata-eval N/A

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

   6. cbrt-prod N/A

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

   7. associate-/r* N/A

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

   8. pow1/3 N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. associate-*l/ N/A

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

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

       \[\leadsto \mathsf{/.f64}\left(\left({2}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{g}\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({2}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(-1 \cdot \frac{1}{3}\right)\right), \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{a}\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{a}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), \mathsf{cbrt.f64}\left(g\right)\right), \left(\sqrt[3]{a}\right)\right) \]

   18. cbrt-lowering-cbrt.f64 98.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

4. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. frac-2neg N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. neg-mul-1 N/A

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

   4. times-frac N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{{2}^{\frac{-1}{3}}}{-1}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(\sqrt[3]{g}\right)}{\sqrt[3]{a}}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({2}^{\frac{-1}{3}}\right), -1\right), \left(\frac{\color{blue}{\mathsf{neg}\left(\sqrt[3]{g}\right)}}{\sqrt[3]{a}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), -1\right), \left(\frac{\mathsf{neg}\left(\color{blue}{\sqrt[3]{g}}\right)}{\sqrt[3]{a}}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), -1\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt[3]{g}\right)\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), -1\right), \mathsf{/.f64}\left(\left(0 - \sqrt[3]{g}\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), -1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{g}\right)\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), -1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(g\right)\right), \left(\sqrt[3]{a}\right)\right)\right) \]

   12. cbrt-lowering-cbrt.f64 98.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{3}\right), -1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(a\right)\right)\right) \]

6. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333}}{-1} \cdot \frac{0 - \sqrt[3]{g}}{\sqrt[3]{a}}} \]
7. Step-by-step derivation

   1. frac-2neg N/A

      \[\leadsto \frac{{2}^{\frac{-1}{3}}}{-1} \cdot \frac{\mathsf{neg}\left(\left(0 - \sqrt[3]{g}\right)\right)}{\color{blue}{\mathsf{neg}\left(\sqrt[3]{a}\right)}} \]

   2. frac-times N/A

      \[\leadsto \frac{{2}^{\frac{-1}{3}} \cdot \left(\mathsf{neg}\left(\left(0 - \sqrt[3]{g}\right)\right)\right)}{\color{blue}{-1 \cdot \left(\mathsf{neg}\left(\sqrt[3]{a}\right)\right)}} \]

   3. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\mathsf{neg}\left({2}^{\frac{-1}{3}} \cdot \left(0 - \sqrt[3]{g}\right)\right)}{\color{blue}{-1} \cdot \left(\mathsf{neg}\left(\sqrt[3]{a}\right)\right)} \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left({2}^{\frac{-1}{3}}\right)\right) \cdot \left(0 - \sqrt[3]{g}\right)}{\color{blue}{-1} \cdot \left(\mathsf{neg}\left(\sqrt[3]{a}\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left(0 - \sqrt[3]{g}\right) \cdot \left(\mathsf{neg}\left({2}^{\frac{-1}{3}}\right)\right)}{\color{blue}{-1} \cdot \left(\mathsf{neg}\left(\sqrt[3]{a}\right)\right)} \]

   6. remove-double-div N/A

      \[\leadsto \frac{\left(0 - \sqrt[3]{g}\right) \cdot \frac{1}{\frac{1}{\mathsf{neg}\left({2}^{\frac{-1}{3}}\right)}}}{-1 \cdot \left(\mathsf{neg}\left(\sqrt[3]{a}\right)\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\left(0 - \sqrt[3]{g}\right) \cdot \frac{1}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left({2}^{\frac{-1}{3}}\right)}}}{-1 \cdot \left(\mathsf{neg}\left(\sqrt[3]{a}\right)\right)} \]

   8. frac-2neg N/A

      \[\leadsto \frac{\left(0 - \sqrt[3]{g}\right) \cdot \frac{1}{\frac{-1}{{2}^{\frac{-1}{3}}}}}{-1 \cdot \left(\mathsf{neg}\left(\sqrt[3]{a}\right)\right)} \]

   9. div-inv N/A

      \[\leadsto \frac{\frac{0 - \sqrt[3]{g}}{\frac{-1}{{2}^{\frac{-1}{3}}}}}{\color{blue}{-1} \cdot \left(\mathsf{neg}\left(\sqrt[3]{a}\right)\right)} \]

   10. sub0-neg N/A

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

   11. div-inv N/A

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

   12. mul-1-neg N/A

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

   13. frac-2neg N/A

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

   14. pow-flip N/A

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

   15. metadata-eval N/A

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

   16. pow1/3 N/A

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

   17. cbrt-div N/A

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

   18. neg-mul-1 N/A

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

   19. remove-double-neg N/A

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

8. Applied egg-rr 98.7%

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

Alternative 4: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. div-inv N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \frac{1}{2 \cdot a}\right)\right) \]

   2. inv-pow N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(2 \cdot a\right)}^{-1}\right)\right) \]

   3. sqr-pow N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \left({\left(2 \cdot a\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(2 \cdot a\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(2 \cdot a\right)}^{-1}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(2 \cdot a\right)}^{\left(2 \cdot \frac{-1}{2}\right)}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(2 \cdot a\right)}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(2 \cdot a\right)}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}\right)\right) \]

   8. pow-pow N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left({\left(2 \cdot a\right)}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

   9. pow2 N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\left(2 \cdot a\right) \cdot \left(2 \cdot a\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

   10. remove-double-neg N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot a\right)\right)\right)\right) \cdot \left(2 \cdot a\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot a\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot a\right)\right)\right)\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

   12. sqr-neg N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\left(\mathsf{neg}\left(2 \cdot a\right)\right) \cdot \left(\mathsf{neg}\left(2 \cdot a\right)\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

   13. pow-prod-down N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \left({\left(\mathsf{neg}\left(2 \cdot a\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot {\left(\mathsf{neg}\left(2 \cdot a\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

   14. pow-prod-up N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\mathsf{neg}\left(2 \cdot a\right)\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\mathsf{neg}\left(2 \cdot a\right)\right)}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\mathsf{neg}\left(2 \cdot a\right)\right)}^{\left(\frac{-1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\mathsf{neg}\left(2 \cdot a\right)\right)}^{\left(\frac{-1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\mathsf{neg}\left(2 \cdot a\right)\right)}^{\left(\frac{-1}{2} + \frac{-1}{2}\right)}\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot {\left(\mathsf{neg}\left(2 \cdot a\right)\right)}^{-1}\right)\right) \]

   20. inv-pow N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(g \cdot \frac{1}{\mathsf{neg}\left(2 \cdot a\right)}\right)\right) \]

4. Applied egg-rr 80.7%

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

5. Final simplification 80.7%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \frac{-1}{\frac{-1}{g}}} \]
6. Add Preprocessing

Alternative 5: 76.4% 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 80.7%

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

   1. clear-num N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{2 \cdot a}{g}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{1}{2 \cdot a} \cdot g\right)\right) \]

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

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), g\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), g\right)\right) \]

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

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2}\right), a\right), g\right)\right) \]

   6. metadata-eval 80.7%

      \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), g\right)\right) \]

4. Applied egg-rr 80.7%

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

5. Final simplification 80.7%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))