2-ancestry mixing, zero discriminant

Percentage Accurate: 76.1% → 98.7%

Time: 3.6s

Alternatives: 6

Speedup: 0.9×

Specification

Math

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

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: 76.1% accurate, 1.0× speedup

Math

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

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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. div-flip N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lower-unsound-/.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. mult-flip N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 98.7%

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

3. Applied rewrites 98.7%

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

Alternative 2: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. mult-flip N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-cbrt.f64 98.7%

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

3. Applied rewrites 98.7%

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

Alternative 3: 98.7% accurate, 0.6× speedup

Math

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

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 76.1%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. div-flip N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lower-unsound-/.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. mult-flip N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 98.7%

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

3. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-flip-rev N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. cbrt-prod N/A

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

   8. associate-/l* N/A

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

   9. lift-cbrt.f64 N/A

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

   10. cbrt-div N/A

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

   11. metadata-eval N/A

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

   12. associate-/r* N/A

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

   13. count-2 N/A

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

   14. lift-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. inv-pow N/A

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

   18. cbrt-pow N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-pow.f64 N/A

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

   22. metadata-eval N/A

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

   23. lower-cbrt.f64 45.9%

       \[\leadsto {\left(a + a\right)}^{-0.3333333333333333} \cdot \color{blue}{\sqrt[3]{g}} \]

5. Applied rewrites 45.9%

   \[\leadsto \color{blue}{{\left(a + a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

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

   4. pow1/3 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2 N/A

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

   7. cbrt-prod N/A

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

   8. lift-cbrt.f64 N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. cbrt-undiv N/A

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

   12. metadata-eval N/A

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

   13. lift-cbrt.f64 N/A

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

   14. cbrt-undiv N/A

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

   15. lift-/.f64 N/A

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

   16. lower-cbrt.f64 98.7%

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

7. Applied rewrites 98.7%

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

Alternative 4: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 98.7%

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

   7. lift-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 98.7%

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

3. Applied rewrites 98.7%

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

Alternative 5: 76.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 76.1%

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

3. Applied rewrites 76.1%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. associate-*l/ N/A

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

   5. lift-+.f64 N/A

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

   6. count-2 N/A

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

   7. frac-times N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. associate-/r/ N/A

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

   12. lower-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. div-flip-rev N/A

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

   15. lower-/.f64 75.5%

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

5. Applied rewrites 75.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r/ N/A

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

   5. associate-*l/ N/A

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

   6. metadata-eval N/A

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

   7. associate-/r/ N/A

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

   8. lift-/.f64 N/A

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

   9. mult-flip-rev N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-times N/A

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. cbrt-div N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. cbrt-prod N/A

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

   20. remove-double-div N/A

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

   21. lift-/.f64 N/A

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

   22. cbrt-prod N/A

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

   23. mult-flip N/A

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

   24. lift-/.f64 N/A

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

7. Applied rewrites 76.0%

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

Alternative 6: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 76.1%

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

3. Applied rewrites 76.1%

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

Reproduce

herbie shell --seed 2025209 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))