2-ancestry mixing, zero discriminant

Percentage Accurate: 76.6% → 98.7%

Time: 6.8s

Alternatives: 8

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: 76.6% 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 76.7%

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -1 \cdot 10^{-304}:\\ \;\;\;\;\sqrt[3]{g \cdot -0.5} \cdot {\left(-a\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -1e-304)
   (* (cbrt (* g -0.5)) (pow (- a) -0.3333333333333333))
   (* (pow a -0.3333333333333333) (cbrt (* g 0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -1e-304) {
		tmp = cbrt((g * -0.5)) * pow(-a, -0.3333333333333333);
	} 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) <= -1e-304) {
		tmp = Math.cbrt((g * -0.5)) * Math.pow(-a, -0.3333333333333333);
	} 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) <= -1e-304)
		tmp = Float64(cbrt(Float64(g * -0.5)) * (Float64(-a) ^ -0.3333333333333333));
	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], -1e-304], N[(N[Power[N[(g * -0.5), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-a), -0.3333333333333333], $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) < -9.99999999999999971e-305

   1. Initial program 74.2%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. cbrt-div N/A

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

      6. div-inv 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. lift-*.f64 N/A

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

      9. cbrt-div N/A

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

      10. frac-2neg N/A

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

      11. div-inv N/A

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

      12. cbrt-prod N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cbrt.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow1/3 N/A

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

      20. inv-pow N/A

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

      21. pow-pow N/A

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

      22. metadata-eval N/A

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

      23. lower-pow.f64 N/A

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

      24. lower-neg.f64 91.8

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

   6. Applied rewrites 91.8%

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

   if -9.99999999999999971e-305 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 79.7%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 92.1

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

   4. Applied rewrites 92.1%

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

Alternative 3: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -1 \cdot 10^{-304}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{g}{a}}{\frac{0.5}{a} \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -1e-304)
   (cbrt (/ (/ g a) (* (/ 0.5 a) (* a 4.0))))
   (* (pow a -0.3333333333333333) (cbrt (* g 0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -1e-304) {
		tmp = cbrt(((g / a) / ((0.5 / a) * (a * 4.0))));
	} 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) <= -1e-304) {
		tmp = Math.cbrt(((g / a) / ((0.5 / a) * (a * 4.0))));
	} 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) <= -1e-304)
		tmp = cbrt(Float64(Float64(g / a) / Float64(Float64(0.5 / a) * Float64(a * 4.0))));
	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], -1e-304], N[Power[N[(N[(g / a), $MachinePrecision] / N[(N[(0.5 / a), $MachinePrecision] * N[(a * 4.0), $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) < -9.99999999999999971e-305

   1. Initial program 74.2%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 74.1

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

   4. Applied rewrites 74.1%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-div N/A

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

      8. unpow1 N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

          \[\leadsto \sqrt[3]{\frac{g}{a} \cdot \left(\frac{a}{a} \cdot \color{blue}{\frac{2}{4}}\right)} \]

      11. times-frac N/A

          \[\leadsto \sqrt[3]{\frac{g}{a} \cdot \color{blue}{\frac{a \cdot 2}{a \cdot 4}}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt[3]{\frac{g}{a} \cdot \frac{\color{blue}{2 \cdot a}}{a \cdot 4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{g}{a} \cdot \frac{\color{blue}{2 \cdot a}}{a \cdot 4}} \]

      14. times-frac N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{g \cdot \left(2 \cdot a\right)}{a \cdot \left(a \cdot 4\right)}}} \]

      15. associate-/r* N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g \cdot \left(2 \cdot a\right)}{a}}{a \cdot 4}}} \]

      16. *-commutative N/A

          \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{\left(2 \cdot a\right) \cdot g}}{a}}{a \cdot 4}} \]

      17. associate-/l* N/A

          \[\leadsto \sqrt[3]{\frac{\color{blue}{\left(2 \cdot a\right) \cdot \frac{g}{a}}}{a \cdot 4}} \]

      18. lift-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{\left(2 \cdot a\right) \cdot \color{blue}{\frac{g}{a}}}{a \cdot 4}} \]

      19. associate-*r/ N/A

          \[\leadsto \sqrt[3]{\color{blue}{\left(2 \cdot a\right) \cdot \frac{\frac{g}{a}}{a \cdot 4}}} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt[3]{\color{blue}{\left(2 \cdot a\right)} \cdot \frac{\frac{g}{a}}{a \cdot 4}} \]

      21. metadata-eval N/A

          \[\leadsto \sqrt[3]{\left(\color{blue}{\frac{1}{\frac{1}{2}}} \cdot a\right) \cdot \frac{\frac{g}{a}}{a \cdot 4}} \]

      22. associate-/r/ N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{\frac{1}{2}}{a}}} \cdot \frac{\frac{g}{a}}{a \cdot 4}} \]

      23. lift-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{\color{blue}{\frac{\frac{1}{2}}{a}}} \cdot \frac{\frac{g}{a}}{a \cdot 4}} \]

      24. frac-times N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{1 \cdot \frac{g}{a}}{\frac{\frac{1}{2}}{a} \cdot \left(a \cdot 4\right)}}} \]

      25. *-lft-identity N/A

          \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{g}{a}}}{\frac{\frac{1}{2}}{a} \cdot \left(a \cdot 4\right)}} \]

      26. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{a}}{\frac{\frac{1}{2}}{a} \cdot \left(a \cdot 4\right)}}} \]

   6. Applied rewrites 74.2%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{a}}{\frac{0.5}{a} \cdot \left(a \cdot 4\right)}}} \]

   if -9.99999999999999971e-305 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 79.7%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 92.1

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

   4. Applied rewrites 92.1%

      \[\leadsto \color{blue}{{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}} \]
3. Recombined 2 regimes into one program.
4. 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 76.7%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 98.8

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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot 0.5}}{\sqrt[3]{a}}} \]
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. associate-/l* N/A

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

   6. unpow-prod-down N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/3 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow1/3 N/A

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

   13. lower-cbrt.f64 98.7

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

6. Applied rewrites 98.7%

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

7. Final simplification 98.7%

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

Alternative 5: 76.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.7%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. cbrt-div N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. clear-num N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. div-inv N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 76.9

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

4. Applied rewrites 76.9%

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

Alternative 6: 76.4% accurate, 0.9× 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 76.7%

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

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

6. Applied rewrites 76.5%

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

   1. associate-*l* N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{4}{\frac{\color{blue}{g \cdot \left(2 \cdot 1\right)}}{a}}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{4}{\frac{g \cdot \color{blue}{2}}{a}}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{4}{\frac{\color{blue}{g \cdot 2}}{a}}}} \]

   4. *-rgt-identity N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{4}{\frac{\color{blue}{\left(g \cdot 2\right) \cdot 1}}{a}}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{4}{\frac{\color{blue}{\left(g \cdot 2\right) \cdot 1}}{a}}}} \]

   6. associate-/r/ N/A

      \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{4}{\left(g \cdot 2\right) \cdot 1} \cdot a}}} \]

   7. clear-num N/A

      \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{1}{\frac{\left(g \cdot 2\right) \cdot 1}{4}}} \cdot a}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{1}{\frac{\color{blue}{\left(g \cdot 2\right) \cdot 1}}{4}} \cdot a}} \]

   9. *-rgt-identity N/A

      \[\leadsto \frac{1}{\sqrt[3]{\frac{1}{\frac{\color{blue}{g \cdot 2}}{4}} \cdot a}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{\sqrt[3]{\frac{1}{\frac{\color{blue}{g \cdot 2}}{4}} \cdot a}} \]

   11. associate-/l* N/A

       \[\leadsto \frac{1}{\sqrt[3]{\frac{1}{\color{blue}{g \cdot \frac{2}{4}}} \cdot a}} \]

   12. metadata-eval N/A

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

   13. associate-/r* N/A

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

   14. associate-/r/ N/A

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

   15. lift-/.f64 N/A

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

   16. *-inverses N/A

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

   17. associate-/r* N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. associate-/l/ N/A

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

   21. lift-/.f64 N/A

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

   22. associate-*l/ N/A

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

   23. associate-/r/ N/A

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

   24. lower-*.f64 N/A

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

8. Applied rewrites 76.8%

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

9. Final simplification 76.8%

   \[\leadsto \frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}} \]
10. Add Preprocessing

Alternative 7: 76.6% 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]

Derivation

1. Initial program 76.7%

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

Alternative 8: 76.6% 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 76.7%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 76.7

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

4. Applied rewrites 76.7%

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

5. Final simplification 76.7%

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

Reproduce

herbie shell --seed 2024209 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))