2-ancestry mixing, zero discriminant

Percentage Accurate: 76.0% → 98.7%

Time: 8.5s

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. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. div-inv N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

6. Applied rewrites 98.8%

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

Alternative 2: 89.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a} \cdot \left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))) (t_1 (/ (cbrt (* a (* a (* g 0.5)))) a)))
   (if (<= t_0 -1e+255)
     t_1
     (if (<= t_0 -2e-302)
       (/ 1.0 (cbrt (/ a (* g 0.5))))
       (if (<= t_0 0.0)
         t_1
         (if (<= t_0 2e+277)
           (cbrt (* (/ g a) (* (sqrt 0.5) (sqrt 0.5))))
           t_1))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = cbrt((a * (a * (g * 0.5)))) / a;
	double tmp;
	if (t_0 <= -1e+255) {
		tmp = t_1;
	} else if (t_0 <= -2e-302) {
		tmp = 1.0 / cbrt((a / (g * 0.5)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+277) {
		tmp = cbrt(((g / a) * (sqrt(0.5) * sqrt(0.5))));
	} 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((a * (a * (g * 0.5)))) / a;
	double tmp;
	if (t_0 <= -1e+255) {
		tmp = t_1;
	} else if (t_0 <= -2e-302) {
		tmp = 1.0 / Math.cbrt((a / (g * 0.5)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+277) {
		tmp = Math.cbrt(((g / a) * (Math.sqrt(0.5) * Math.sqrt(0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(cbrt(Float64(a * Float64(a * Float64(g * 0.5)))) / a)
	tmp = 0.0
	if (t_0 <= -1e+255)
		tmp = t_1;
	elseif (t_0 <= -2e-302)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+277)
		tmp = cbrt(Float64(Float64(g / a) * Float64(sqrt(0.5) * sqrt(0.5))));
	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[(a * N[(a * N[(g * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+255], t$95$1, If[LessEqual[t$95$0, -2e-302], N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+277], N[Power[N[(N[(g / a), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -9.99999999999999988e254 or -1.9999999999999999e-302 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 0.0 or 2.00000000000000001e277 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 18.8%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 98.6

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

   4. Applied rewrites 98.6%

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

   6. Applied rewrites 98.6%

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

   8. Applied rewrites 66.2%

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

   if -9.99999999999999988e254 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -1.9999999999999999e-302

   1. Initial program 99.1%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. clear-num N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

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

   if 0.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 2.00000000000000001e277

   1. Initial program 98.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 97.3

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

   4. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. sqrt-pow1 N/A

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

      6. inv-pow N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. sqrt-prod N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. *-commutative N/A

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

      16. sqrt-pow1 N/A

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

      17. inv-pow N/A

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

      18. clear-num N/A

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

      19. div-inv N/A

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

      20. lift-/.f64 N/A

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

      21. clear-num N/A

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

      22. lift-/.f64 N/A

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

      23. sqrt-prod N/A

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

      24. lift-sqrt.f64 N/A

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

      25. lift-sqrt.f64 N/A

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

      26. *-commutative N/A

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

   6. Applied rewrites 98.3%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -1 \cdot 10^{+255}:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a} \cdot \left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Reproduce

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