2-ancestry mixing, zero discriminant

Percentage Accurate: 76.4% → 98.7%

Time: 7.2s

Alternatives: 7

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: 76.4% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. associate-/r* N/A

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

   2. 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 egg-rr 98.7%

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

Alternative 2: 92.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.9%

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

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

      4. neg-mul-1 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. cbrt-prod N/A

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

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

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

      9. times-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. cbrt-div N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-cbrt.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-cbrt.f64 N/A

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

      17. div-inv N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 98.8

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

   4. Applied egg-rr 98.8%

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

   6. Applied egg-rr 92.0%

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

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

   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. div-inv N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 92.1

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

   4. Applied egg-rr 92.1%

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

4. Final simplification 92.0%

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

Alternative 3: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 10^{-278}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(a \cdot 2\right)}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) 1e-278)
   (/ 1.0 (cbrt (* a (/ 2.0 g))))
   (* (cbrt g) (pow (* a 2.0) -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 1e-278) {
		tmp = 1.0 / cbrt((a * (2.0 / g)));
	} else {
		tmp = cbrt(g) * pow((a * 2.0), -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 1e-278) {
		tmp = 1.0 / Math.cbrt((a * (2.0 / g)));
	} else {
		tmp = Math.cbrt(g) * Math.pow((a * 2.0), -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= 1e-278)
		tmp = Float64(1.0 / cbrt(Float64(a * Float64(2.0 / g))));
	else
		tmp = Float64(cbrt(g) * (Float64(a * 2.0) ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], 1e-278], N[(1.0 / N[Power[N[(a * N[(2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(a * 2.0), $MachinePrecision], -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < 9.99999999999999938e-279

   1. Initial program 74.5%

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

      1. lift-*.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. sqr-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-pow N/A

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

      10. pow2 N/A

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

      11. remove-double-neg N/A

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

      12. remove-double-neg N/A

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

      13. sqr-neg N/A

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

      14. pow-prod-down N/A

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

      15. pow-prod-up N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. inv-pow N/A

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

   4. Applied egg-rr 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. associate-/r* N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

      14. clear-num N/A

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

      15. cbrt-div N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

   6. Applied egg-rr 37.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. associate-/l/ N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/l/ N/A

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

      18. associate-/r* N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 74.6

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

   8. Applied egg-rr 74.6%

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

   if 9.99999999999999938e-279 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 73.6%

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

      1. lift-*.f64 N/A

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

      2. div-inv N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 92.1

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

   4. Applied egg-rr 92.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 10^{-278}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(a \cdot 2\right)}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 10^{-278}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) 1e-278)
   (/ 1.0 (cbrt (* a (/ 2.0 g))))
   (* (cbrt (* g 0.5)) (pow a -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 1e-278) {
		tmp = 1.0 / cbrt((a * (2.0 / g)));
	} else {
		tmp = cbrt((g * 0.5)) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 1e-278) {
		tmp = 1.0 / Math.cbrt((a * (2.0 / g)));
	} else {
		tmp = Math.cbrt((g * 0.5)) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= 1e-278)
		tmp = Float64(1.0 / cbrt(Float64(a * Float64(2.0 / g))));
	else
		tmp = Float64(cbrt(Float64(g * 0.5)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], 1e-278], N[(1.0 / N[Power[N[(a * N[(2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < 9.99999999999999938e-279

   1. Initial program 74.5%

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

      1. lift-*.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. sqr-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-pow N/A

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

      10. pow2 N/A

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

      11. remove-double-neg N/A

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

      12. remove-double-neg N/A

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

      13. sqr-neg N/A

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

      14. pow-prod-down N/A

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

      15. pow-prod-up N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. inv-pow N/A

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

   4. Applied egg-rr 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. associate-/r* N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

      14. clear-num N/A

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

      15. cbrt-div N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

   6. Applied egg-rr 37.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. associate-/l/ N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/l/ N/A

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

      18. associate-/r* N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 74.6

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

   8. Applied egg-rr 74.6%

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

   if 9.99999999999999938e-279 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 73.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 egg-rr 92.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 10^{-278}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.0%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 98.7

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

4. Applied egg-rr 98.7%

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

5. Final simplification 98.7%

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

Alternative 6: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

3. Final simplification 74.0%

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

Alternative 7: 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 74.0%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 74.0

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

4. Applied egg-rr 74.0%

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

5. Final simplification 74.0%

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

Reproduce

herbie shell --seed 2024211 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))