2-ancestry mixing, zero discriminant

Percentage Accurate: 76.0% → 98.7%

Time: 6.6s

Alternatives: 9

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

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

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(Float64(-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 79.5%

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

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

6. Applied egg-rr 98.8%

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

Alternative 2: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-294}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {\left(a \cdot -2\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) -4e-294)
   (* (cbrt (- g)) (pow (* a -2.0) -0.3333333333333333))
   (* (cbrt (* g 0.5)) (pow a -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-294) {
		tmp = cbrt(-g) * pow((a * -2.0), -0.3333333333333333);
	} 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) <= -4e-294) {
		tmp = Math.cbrt(-g) * Math.pow((a * -2.0), -0.3333333333333333);
	} 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) <= -4e-294)
		tmp = Float64(cbrt(Float64(-g)) * (Float64(a * -2.0) ^ -0.3333333333333333));
	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], -4e-294], N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[N[(a * -2.0), $MachinePrecision], -0.3333333333333333], $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) < -4.00000000000000007e-294

   1. Initial program 78.2%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 egg-rr 98.8%

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

   6. Applied egg-rr 98.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.9

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

      7. lift-cbrt.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. inv-pow N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

      16. lift-*.f64 N/A

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

      17. pow-pow N/A

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

      18. lower-pow.f64 N/A

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

      19. lift-*.f64 N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval 92.3

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

   8. Applied egg-rr 92.3%

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

   if -4.00000000000000007e-294 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 80.7%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 92.4

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

   4. Applied egg-rr 92.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-294}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {\left(a \cdot -2\right)}^{-0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 4 \cdot 10^{-276}:\\ \;\;\;\;\sqrt[3]{\left(g \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{0.5}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) 4e-276)
   (cbrt (* (* g (sqrt 0.5)) (/ (sqrt 0.5) a)))
   (* (cbrt (* g 0.5)) (pow a -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 4e-276) {
		tmp = cbrt(((g * sqrt(0.5)) * (sqrt(0.5) / a)));
	} 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) <= 4e-276) {
		tmp = Math.cbrt(((g * Math.sqrt(0.5)) * (Math.sqrt(0.5) / a)));
	} 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) <= 4e-276)
		tmp = cbrt(Float64(Float64(g * sqrt(0.5)) * Float64(sqrt(0.5) / a)));
	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], 4e-276], N[Power[N[(N[(g * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], 1/3], $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) < 4e-276

   1. Initial program 78.7%

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

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 78.7

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

   4. Applied egg-rr 78.7%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   6. Applied egg-rr 78.7%

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

   if 4e-276 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 80.3%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 92.5

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

   4. Applied egg-rr 92.5%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 4 \cdot 10^{-276}:\\ \;\;\;\;\sqrt[3]{\left(g \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{0.5}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 79.5%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 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 79.5%

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

   1. lift-*.f64 N/A

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

   2. 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: 78.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{\sqrt[3]{0.5 \cdot \left(g \cdot \left(a \cdot a\right)\right)}}{a}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\sqrt[3]{\left(g \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{0.5}}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))) (t_1 (/ (cbrt (* 0.5 (* g (* a a)))) a)))
   (if (<= t_0 -2e+290)
     t_1
     (if (<= t_0 2e+277) (cbrt (* (* g (sqrt 0.5)) (/ (sqrt 0.5) a))) t_1))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = cbrt((0.5 * (g * (a * a)))) / a;
	double tmp;
	if (t_0 <= -2e+290) {
		tmp = t_1;
	} else if (t_0 <= 2e+277) {
		tmp = cbrt(((g * sqrt(0.5)) * (sqrt(0.5) / a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = Math.cbrt((0.5 * (g * (a * a)))) / a;
	double tmp;
	if (t_0 <= -2e+290) {
		tmp = t_1;
	} else if (t_0 <= 2e+277) {
		tmp = Math.cbrt(((g * Math.sqrt(0.5)) * (Math.sqrt(0.5) / a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(cbrt(Float64(0.5 * Float64(g * Float64(a * a)))) / a)
	tmp = 0.0
	if (t_0 <= -2e+290)
		tmp = t_1;
	elseif (t_0 <= 2e+277)
		tmp = cbrt(Float64(Float64(g * sqrt(0.5)) * Float64(sqrt(0.5) / a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(0.5 * N[(g * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+290], t$95$1, If[LessEqual[t$95$0, 2e+277], N[Power[N[(N[(g * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -2.00000000000000012e290 or 2.00000000000000001e277 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 10.7%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 98.2

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

   4. Applied egg-rr 98.2%

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

   6. Applied egg-rr 98.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cbrt-unprod N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r/ N/A

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

      16. inv-pow N/A

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

      17. metadata-eval N/A

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

      18. pow-div N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. cube-unmult N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*l/ N/A

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

   8. Applied egg-rr 39.2%

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

   if -2.00000000000000012e290 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 2.00000000000000001e277

   1. Initial program 89.0%

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

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 88.5

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

   4. Applied egg-rr 88.5%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   6. Applied egg-rr 89.0%

      \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{0.5} \cdot g\right) \cdot \frac{\sqrt{0.5}}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -2 \cdot 10^{+290}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(g \cdot \left(a \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\sqrt[3]{\left(g \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{0.5}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(g \cdot \left(a \cdot a\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{\sqrt[3]{0.5 \cdot \left(g \cdot \left(a \cdot a\right)\right)}}{a}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\sqrt[3]{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))) (t_1 (/ (cbrt (* 0.5 (* g (* a a)))) a)))
   (if (<= t_0 (- INFINITY)) t_1 (if (<= t_0 2e+277) (cbrt t_0) t_1))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = cbrt((0.5 * (g * (a * a)))) / a;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 2e+277) {
		tmp = cbrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = Math.cbrt((0.5 * (g * (a * a)))) / a;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 2e+277) {
		tmp = Math.cbrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(cbrt(Float64(0.5 * Float64(g * Float64(a * a)))) / a)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 2e+277)
		tmp = cbrt(t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(0.5 * N[(g * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 2e+277], N[Power[t$95$0, 1/3], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -inf.0 or 2.00000000000000001e277 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 7.7%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 98.2

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

   4. Applied egg-rr 98.2%

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

   6. Applied egg-rr 98.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cbrt-unprod N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lift-neg.f64 N/A

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

      10. frac-2neg N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r/ N/A

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

      16. inv-pow N/A

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

      17. metadata-eval N/A

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

      18. pow-div N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. cube-unmult N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*l/ N/A

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

   8. Applied egg-rr 37.2%

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

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 2.00000000000000001e277

   1. Initial program 89.0%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(g \cdot \left(a \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(g \cdot \left(a \cdot a\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.0% 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 79.5%

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

3. Final simplification 79.5%

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

Alternative 9: 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 79.5%

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

   1. lift-*.f64 N/A

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

   2. 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 79.5

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

4. Applied egg-rr 79.5%

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

5. Final simplification 79.5%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))