2-ancestry mixing, zero discriminant

Percentage Accurate: 76.0% → 98.7%

Time: 7.1s

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: 76.0% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. lift-*.f64 N/A

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

   2. cbrt-div N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-cbrt.f64 98.8

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

4. Applied rewrites 98.8%

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

Alternative 2: 91.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -5 \cdot 10^{-307}:\\ \;\;\;\;{\left(a \cdot -2\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-g}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -5e-307)
   (* (pow (* a -2.0) -0.3333333333333333) (cbrt (- g)))
   (* (pow a -0.3333333333333333) (cbrt (* g 0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-307) {
		tmp = pow((a * -2.0), -0.3333333333333333) * cbrt(-g);
	} else {
		tmp = pow(a, -0.3333333333333333) * cbrt((g * 0.5));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-307) {
		tmp = Math.pow((a * -2.0), -0.3333333333333333) * Math.cbrt(-g);
	} else {
		tmp = Math.pow(a, -0.3333333333333333) * Math.cbrt((g * 0.5));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -5e-307)
		tmp = Float64((Float64(a * -2.0) ^ -0.3333333333333333) * cbrt(Float64(-g)));
	else
		tmp = Float64((a ^ -0.3333333333333333) * cbrt(Float64(g * 0.5)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -5e-307], N[(N[Power[N[(a * -2.0), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[(-g), 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -5.00000000000000014e-307

   1. Initial program 77.3%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-cbrt.f64 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. cbrt-undiv N/A

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

      3. pow1/3 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. neg-mul-1 N/A

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

      12. lift-neg.f64 N/A

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

      13. unpow-prod-down N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. inv-pow N/A

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

      18. div-inv N/A

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

      19. metadata-eval N/A

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

      20. lift-*.f64 N/A

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

      21. pow-pow N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-pow.f64 N/A

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

      25. metadata-eval N/A

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

      26. pow1/3 N/A

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

      27. lower-cbrt.f64 92.2

          \[\leadsto {\left(a \cdot -2\right)}^{-0.3333333333333333} \cdot \color{blue}{\sqrt[3]{-g}} \]

   6. Applied rewrites 92.2%

      \[\leadsto \color{blue}{{\left(a \cdot -2\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-g}} \]

   if -5.00000000000000014e-307 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 78.3%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 92.2

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

   4. Applied rewrites 92.2%

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

Alternative 3: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 5 \cdot 10^{-301}:\\ \;\;\;\;\sqrt[3]{-\frac{g}{a} \cdot \left(g \cdot \frac{-0.5}{g}\right)}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) 5e-301)
   (cbrt (- (* (/ g a) (* g (/ -0.5 g)))))
   (* (pow a -0.3333333333333333) (cbrt (* g 0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= 5e-301) {
		tmp = cbrt(-((g / a) * (g * (-0.5 / g))));
	} else {
		tmp = pow(a, -0.3333333333333333) * cbrt((g * 0.5));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= 5e-301) {
		tmp = Math.cbrt(-((g / a) * (g * (-0.5 / g))));
	} else {
		tmp = Math.pow(a, -0.3333333333333333) * Math.cbrt((g * 0.5));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= 5e-301)
		tmp = cbrt(Float64(-Float64(Float64(g / a) * Float64(g * Float64(-0.5 / g)))));
	else
		tmp = Float64((a ^ -0.3333333333333333) * cbrt(Float64(g * 0.5)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], 5e-301], N[Power[(-N[(N[(g / a), $MachinePrecision] * N[(g * N[(-0.5 / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), 1/3], $MachinePrecision], N[(N[Power[a, -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < 5.00000000000000013e-301

   1. Initial program 77.6%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 77.6

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

   4. Applied rewrites 77.6%

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. neg-mul-1 N/A

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

      9. lift-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 44.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-neg.f64 N/A

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

      18. neg-mul-1 N/A

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

      19. associate-*r* N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 44.4

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

   6. Applied rewrites 44.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-neg-out N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l/ N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. lower-*.f64 77.6

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

   8. Applied rewrites 77.6%

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

   if 5.00000000000000013e-301 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 78.0%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 92.2

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

   4. Applied rewrites 92.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 5 \cdot 10^{-301}:\\ \;\;\;\;\sqrt[3]{-\frac{g}{a} \cdot \left(g \cdot \frac{-0.5}{g}\right)}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 77.8%

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

   1. lift-*.f64 N/A

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

   2. cbrt-div N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-cbrt.f64 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. cbrt-undiv N/A

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

   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. div-inv N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. associate-*r* N/A

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

   10. neg-mul-1 N/A

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

   11. associate-*l/ N/A

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

   12. cbrt-prod N/A

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

   13. neg-mul-1 N/A

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

   14. cbrt-prod N/A

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

   15. *-rgt-identity N/A

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

   16. *-inverses N/A

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

   17. associate-/l* N/A

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

   18. lift-*.f64 N/A

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

   19. cbrt-prod N/A

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

   20. associate-/l* N/A

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

   21. neg-mul-1 N/A

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

   22. lift-neg.f64 N/A

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

6. Applied rewrites 98.7%

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

   1. associate-*l/ N/A

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

   2. metadata-eval N/A

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

   3. lower-/.f64 98.7

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

8. Applied rewrites 98.7%

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

9. Final simplification 98.7%

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

Alternative 5: 76.0% 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 77.8%

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

   1. lift-*.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 77.8

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

4. Applied rewrites 77.8%

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

5. Final simplification 77.8%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))