2-ancestry mixing, zero discriminant

Percentage Accurate: 77.0% → 98.7%

Time: 8.7s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 77.0% 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(g) * cbrt(Float64(0.5 / a)))
end

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. cbrt-prod N/A

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

   4. cbrt-div N/A

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

   5. metadata-eval N/A

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

   6. cbrt-prod N/A

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

   7. associate-/r* N/A

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

   8. pow1/3 N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. associate-*l/ N/A

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

   13. /-lowering-/.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. pow-lowering-pow.f64 N/A

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

   16. metadata-eval N/A

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

   17. cbrt-lowering-cbrt.f64 N/A

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

   18. cbrt-lowering-cbrt.f64 98.5

       \[\leadsto \frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. div-inv N/A

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

   4. pow1/3 N/A

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

   5. pow-flip N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-down N/A

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

   8. metadata-eval N/A

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

   9. associate-/r/ N/A

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

   10. metadata-eval N/A

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

   11. pow-pow N/A

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

   12. inv-pow N/A

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

   13. associate-/r/ N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

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

   16. pow1/3 N/A

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

   17. cbrt-div N/A

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

   18. unpow1/3 N/A

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

   19. frac-2neg N/A

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

   20. associate-/l* N/A

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

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

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

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

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

6. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\left(0 - \sqrt[3]{g}\right) \cdot \frac{\sqrt[3]{0.5}}{0 - \sqrt[3]{a}}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

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

   2. distribute-frac-neg2 N/A

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

   3. neg-lowering-neg.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. +-rgt-identity N/A

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

   10. cbrt-lowering-cbrt.f64 N/A

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

   11. +-rgt-identity N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. clear-num N/A

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

   16. /-lowering-/.f64 98.7

       \[\leadsto \left(0 - \sqrt[3]{g}\right) \cdot \left(-\sqrt[3]{\color{blue}{\frac{0.5}{a}}}\right) \]

8. Applied egg-rr 98.7%

   \[\leadsto \left(0 - \sqrt[3]{g}\right) \cdot \color{blue}{\left(-\sqrt[3]{\frac{0.5}{a}}\right)} \]
9. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. cbrt-lowering-cbrt.f64 98.7

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

10. Applied egg-rr 98.7%

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

11. Final simplification 98.7%

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

Alternative 2: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \mathsf{fma}\left(g, 0.25 \cdot \left(g \cdot a\right), 0\right)\\ t_2 := \frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{t\_1}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(g, 0.5, 0\right) \cdot \frac{1}{\frac{1}{\sqrt[3]{\frac{1}{t\_1}}}}\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{\mathsf{fma}\left(g, 0.5, 0\right)}}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0)))
        (t_1 (fma g (* 0.25 (* g a)) 0.0))
        (t_2 (/ (fma g 0.5 0.0) (cbrt t_1))))
   (if (<= t_0 (- INFINITY))
     (* (fma g 0.5 0.0) (/ 1.0 (/ 1.0 (cbrt (/ 1.0 t_1)))))
     (if (<= t_0 -4e-318)
       (cbrt (* g (/ 0.5 a)))
       (if (<= t_0 2e-309)
         t_2
         (if (<= t_0 4e+307) (/ 1.0 (cbrt (/ a (fma g 0.5 0.0)))) t_2))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = fma(g, (0.25 * (g * a)), 0.0);
	double t_2 = fma(g, 0.5, 0.0) / cbrt(t_1);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(g, 0.5, 0.0) * (1.0 / (1.0 / cbrt((1.0 / t_1))));
	} else if (t_0 <= -4e-318) {
		tmp = cbrt((g * (0.5 / a)));
	} else if (t_0 <= 2e-309) {
		tmp = t_2;
	} else if (t_0 <= 4e+307) {
		tmp = 1.0 / cbrt((a / fma(g, 0.5, 0.0)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = fma(g, Float64(0.25 * Float64(g * a)), 0.0)
	t_2 = Float64(fma(g, 0.5, 0.0) / cbrt(t_1))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(g, 0.5, 0.0) * Float64(1.0 / Float64(1.0 / cbrt(Float64(1.0 / t_1)))));
	elseif (t_0 <= -4e-318)
		tmp = cbrt(Float64(g * Float64(0.5 / a)));
	elseif (t_0 <= 2e-309)
		tmp = t_2;
	elseif (t_0 <= 4e+307)
		tmp = Float64(1.0 / cbrt(Float64(a / fma(g, 0.5, 0.0))));
	else
		tmp = t_2;
	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[(g * N[(0.25 * N[(g * a), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(g * 0.5 + 0.0), $MachinePrecision] / N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(g * 0.5 + 0.0), $MachinePrecision] * N[(1.0 / N[(1.0 / N[Power[N[(1.0 / t$95$1), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -4e-318], N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t$95$0, 2e-309], t$95$2, If[LessEqual[t$95$0, 4e+307], N[(1.0 / N[Power[N[(a / N[(g * 0.5 + 0.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 4.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. cbrt-prod N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. cbrt-prod N/A

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

      7. associate-/r* N/A

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

      8. pow1/3 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. pow-lowering-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. cbrt-lowering-cbrt.f64 N/A

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

      18. cbrt-lowering-cbrt.f64 97.8

          \[\leadsto \frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

   4. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]

   6. Applied egg-rr 19.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(g, 0.5, 0\right) \cdot \frac{1}{\sqrt[3]{a \cdot \mathsf{fma}\left(0.25, g \cdot g, 0\right)}}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{a \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(g \cdot g\right)\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\left(\frac{1}{4} \cdot \left(g \cdot g\right)\right) \cdot a}}} \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\left(\left(\frac{1}{4} \cdot g\right) \cdot g\right)} \cdot a}} \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\left(\frac{1}{4} \cdot g\right) \cdot \left(g \cdot a\right)}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\left(\frac{1}{4} \cdot g\right) \cdot \left(g \cdot a\right)}}} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\left(g \cdot \frac{1}{4}\right)} \cdot \left(g \cdot a\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\left(g \cdot \frac{1}{4}\right)} \cdot \left(g \cdot a\right)}} \]

      8. *-lowering-*.f64 64.6

         \[\leadsto \mathsf{fma}\left(g, 0.5, 0\right) \cdot \frac{1}{\sqrt[3]{\left(g \cdot 0.25\right) \cdot \color{blue}{\left(g \cdot a\right)}}} \]

   8. Applied egg-rr 64.6%

      \[\leadsto \mathsf{fma}\left(g, 0.5, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\left(g \cdot 0.25\right) \cdot \left(g \cdot a\right)}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\left(g \cdot a\right) \cdot \left(g \cdot \frac{1}{4}\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\left(g \cdot \frac{1}{4}\right) \cdot \left(g \cdot a\right)}}} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\left(\frac{1}{4} \cdot g\right)} \cdot \left(g \cdot a\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\frac{1}{4} \cdot \left(g \cdot \left(g \cdot a\right)\right)}}} \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\frac{1}{4} \cdot \color{blue}{\left(\left(g \cdot g\right) \cdot a\right)}}} \]

      6. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\frac{1}{4} \cdot \left(\color{blue}{\left(g \cdot g + 0\right)} \cdot a\right)}} \]

      7. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right) + 0}}} \]

      8. flip-+ N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\frac{\left(\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right)\right) \cdot \left(\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right)\right) - 0 \cdot 0}{\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right) - 0}}}} \]

      9. clear-num N/A

         \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{\frac{1}{\frac{\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right) - 0}{\left(\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right)\right) \cdot \left(\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right)\right) - 0 \cdot 0}}}}} \]

      10. cbrt-div N/A

          \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right) - 0}{\left(\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right)\right) \cdot \left(\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right)\right) - 0 \cdot 0}}}}} \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\frac{\color{blue}{1}}{\sqrt[3]{\frac{\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right) - 0}{\left(\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right)\right) \cdot \left(\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right)\right) - 0 \cdot 0}}}} \]

      12. --rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(g, \frac{1}{2}, 0\right) \cdot \frac{1}{\frac{1}{\sqrt[3]{\frac{\color{blue}{\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right)}}{\left(\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right)\right) \cdot \left(\frac{1}{4} \cdot \left(\left(g \cdot g + 0\right) \cdot a\right)\right) - 0 \cdot 0}}}} \]

   10. Applied egg-rr 65.0%

       \[\leadsto \mathsf{fma}\left(g, 0.5, 0\right) \cdot \frac{1}{\color{blue}{\frac{1}{\sqrt[3]{\frac{1}{\mathsf{fma}\left(g, 0.25 \cdot \left(g \cdot a\right), 0\right)}}}}} \]

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -3.9999999e-318

   1. Initial program 98.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

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

      6. metadata-eval 98.7

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

   4. Applied egg-rr 98.7%

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

   if -3.9999999e-318 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 1.9999999999999988e-309 or 3.99999999999999994e307 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 6.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. cbrt-prod N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. cbrt-prod N/A

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

      7. associate-/r* N/A

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

      8. pow1/3 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. pow-lowering-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. cbrt-lowering-cbrt.f64 N/A

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

      18. cbrt-lowering-cbrt.f64 98.4

          \[\leadsto \frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

   4. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-inv N/A

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

      4. pow1/3 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-down N/A

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

      8. metadata-eval N/A

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

      9. associate-/r/ N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. inv-pow N/A

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

      16. pow1/3 N/A

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

      17. cbrt-div N/A

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

      18. unpow1/3 N/A

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

      19. frac-2neg N/A

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

      20. associate-/l* N/A

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

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

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

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

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

   6. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\left(0 - \sqrt[3]{g}\right) \cdot \frac{\sqrt[3]{0.5}}{0 - \sqrt[3]{a}}} \]
   7. Step-by-step derivation

      1. sub0-neg N/A

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

      2. distribute-frac-neg2 N/A

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

      3. neg-lowering-neg.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. +-rgt-identity N/A

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

      10. cbrt-lowering-cbrt.f64 N/A

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

      11. +-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. clear-num N/A

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

      16. /-lowering-/.f64 98.4

          \[\leadsto \left(0 - \sqrt[3]{g}\right) \cdot \left(-\sqrt[3]{\color{blue}{\frac{0.5}{a}}}\right) \]

   8. Applied egg-rr 98.4%

      \[\leadsto \left(0 - \sqrt[3]{g}\right) \cdot \color{blue}{\left(-\sqrt[3]{\frac{0.5}{a}}\right)} \]

   10. Applied egg-rr 71.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{\mathsf{fma}\left(g, 0.25 \cdot \left(g \cdot a\right), 0\right)}}} \]

   if 1.9999999999999988e-309 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 3.99999999999999994e307

   1. Initial program 99.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. cbrt-prod N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. cbrt-prod N/A

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

      7. associate-/r* N/A

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

      8. pow1/3 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. pow-lowering-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. cbrt-lowering-cbrt.f64 N/A

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

      18. cbrt-lowering-cbrt.f64 98.6

          \[\leadsto \frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

   4. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]

   6. Applied egg-rr 92.6%

      \[\leadsto \color{blue}{{\left(\frac{a}{\mathsf{fma}\left(g, 0.5, 0\right)}\right)}^{-0.3333333333333333}} \]

   8. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{a}{\mathsf{fma}\left(g, 0.5, 0\right)}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(g, 0.5, 0\right) \cdot \frac{1}{\frac{1}{\sqrt[3]{\frac{1}{\mathsf{fma}\left(g, 0.25 \cdot \left(g \cdot a\right), 0\right)}}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{\mathsf{fma}\left(g, 0.25 \cdot \left(g \cdot a\right), 0\right)}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{\mathsf{fma}\left(g, 0.5, 0\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{\mathsf{fma}\left(g, 0.25 \cdot \left(g \cdot a\right), 0\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{\mathsf{fma}\left(g, 0.25 \cdot \left(g \cdot a\right), 0\right)}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{\mathsf{fma}\left(g, 0.5, 0\right)}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0)))
        (t_1 (/ (fma g 0.5 0.0) (cbrt (fma g (* 0.25 (* g a)) 0.0)))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -4e-318)
       (cbrt (* g (/ 0.5 a)))
       (if (<= t_0 2e-309)
         t_1
         (if (<= t_0 4e+307) (/ 1.0 (cbrt (/ a (fma g 0.5 0.0)))) t_1))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = fma(g, 0.5, 0.0) / cbrt(fma(g, (0.25 * (g * a)), 0.0));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -4e-318) {
		tmp = cbrt((g * (0.5 / a)));
	} else if (t_0 <= 2e-309) {
		tmp = t_1;
	} else if (t_0 <= 4e+307) {
		tmp = 1.0 / cbrt((a / fma(g, 0.5, 0.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(fma(g, 0.5, 0.0) / cbrt(fma(g, Float64(0.25 * Float64(g * a)), 0.0)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -4e-318)
		tmp = cbrt(Float64(g * Float64(0.5 / a)));
	elseif (t_0 <= 2e-309)
		tmp = t_1;
	elseif (t_0 <= 4e+307)
		tmp = Float64(1.0 / cbrt(Float64(a / fma(g, 0.5, 0.0))));
	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[(g * 0.5 + 0.0), $MachinePrecision] / N[Power[N[(g * N[(0.25 * N[(g * a), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -4e-318], N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t$95$0, 2e-309], t$95$1, If[LessEqual[t$95$0, 4e+307], N[(1.0 / N[Power[N[(a / N[(g * 0.5 + 0.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -inf.0 or -3.9999999e-318 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 1.9999999999999988e-309 or 3.99999999999999994e307 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 6.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. cbrt-prod N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. cbrt-prod N/A

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

      7. associate-/r* N/A

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

      8. pow1/3 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. pow-lowering-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. cbrt-lowering-cbrt.f64 N/A

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

      18. cbrt-lowering-cbrt.f64 98.2

          \[\leadsto \frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-inv N/A

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

      4. pow1/3 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-down N/A

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

      8. metadata-eval N/A

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

      9. associate-/r/ N/A

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

      10. metadata-eval N/A

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

      11. pow-pow N/A

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

      12. inv-pow N/A

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

      16. pow1/3 N/A

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

      17. cbrt-div N/A

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

      18. unpow1/3 N/A

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

      19. frac-2neg N/A

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

      20. associate-/l* N/A

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

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

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

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

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

   6. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{\left(0 - \sqrt[3]{g}\right) \cdot \frac{\sqrt[3]{0.5}}{0 - \sqrt[3]{a}}} \]
   7. Step-by-step derivation

      1. sub0-neg N/A

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

      2. distribute-frac-neg2 N/A

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

      3. neg-lowering-neg.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. +-rgt-identity N/A

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

      10. cbrt-lowering-cbrt.f64 N/A

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

      11. +-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. clear-num N/A

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

      16. /-lowering-/.f64 98.5

          \[\leadsto \left(0 - \sqrt[3]{g}\right) \cdot \left(-\sqrt[3]{\color{blue}{\frac{0.5}{a}}}\right) \]

   8. Applied egg-rr 98.5%

      \[\leadsto \left(0 - \sqrt[3]{g}\right) \cdot \color{blue}{\left(-\sqrt[3]{\frac{0.5}{a}}\right)} \]

   10. Applied egg-rr 69.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{\mathsf{fma}\left(g, 0.25 \cdot \left(g \cdot a\right), 0\right)}}} \]

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -3.9999999e-318

   1. Initial program 98.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

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

      6. metadata-eval 98.7

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

   4. Applied egg-rr 98.7%

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

   if 1.9999999999999988e-309 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 3.99999999999999994e307

   1. Initial program 99.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. cbrt-prod N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. cbrt-prod N/A

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

      7. associate-/r* N/A

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

      8. pow1/3 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. pow-lowering-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. cbrt-lowering-cbrt.f64 N/A

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

      18. cbrt-lowering-cbrt.f64 98.6

          \[\leadsto \frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

   4. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]

   6. Applied egg-rr 92.6%

      \[\leadsto \color{blue}{{\left(\frac{a}{\mathsf{fma}\left(g, 0.5, 0\right)}\right)}^{-0.3333333333333333}} \]

   8. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{a}{\mathsf{fma}\left(g, 0.5, 0\right)}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{\mathsf{fma}\left(g, 0.25 \cdot \left(g \cdot a\right), 0\right)}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{\mathsf{fma}\left(g, 0.25 \cdot \left(g \cdot a\right), 0\right)}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{\mathsf{fma}\left(g, 0.5, 0\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{\mathsf{fma}\left(g, 0.25 \cdot \left(g \cdot a\right), 0\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 76.4%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. cbrt-prod N/A

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

   4. cbrt-div N/A

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

   5. metadata-eval N/A

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

   6. cbrt-prod N/A

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

   7. associate-/r* N/A

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

   8. pow1/3 N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. associate-*l/ N/A

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

   13. /-lowering-/.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. pow-lowering-pow.f64 N/A

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

   16. metadata-eval N/A

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

   17. cbrt-lowering-cbrt.f64 N/A

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

   18. cbrt-lowering-cbrt.f64 98.5

       \[\leadsto \frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]
5. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. div-inv N/A

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

   4. pow1/3 N/A

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

   5. pow-flip N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-down N/A

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

   8. metadata-eval N/A

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

   9. associate-/r/ N/A

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

   10. metadata-eval N/A

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

   11. pow-pow N/A

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

   12. inv-pow N/A

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

   13. associate-/r/ N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

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

   16. pow1/3 N/A

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

   17. cbrt-div N/A

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

   18. unpow1/3 N/A

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

   19. frac-2neg N/A

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

   20. associate-/l* N/A

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

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

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

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

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

6. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\left(0 - \sqrt[3]{g}\right) \cdot \frac{\sqrt[3]{0.5}}{0 - \sqrt[3]{a}}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

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

   2. distribute-frac-neg2 N/A

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

   3. neg-lowering-neg.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. +-rgt-identity N/A

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

   10. cbrt-lowering-cbrt.f64 N/A

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

   11. +-rgt-identity N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. clear-num N/A

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

   16. /-lowering-/.f64 98.7

       \[\leadsto \left(0 - \sqrt[3]{g}\right) \cdot \left(-\sqrt[3]{\color{blue}{\frac{0.5}{a}}}\right) \]

8. Applied egg-rr 98.7%

   \[\leadsto \left(0 - \sqrt[3]{g}\right) \cdot \color{blue}{\left(-\sqrt[3]{\frac{0.5}{a}}\right)} \]
9. Step-by-step derivation

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

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

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

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

   3. clear-num N/A

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

   4. cbrt-div N/A

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

   5. metadata-eval N/A

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

   6. un-div-inv N/A

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

   7. sub0-neg N/A

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

   8. remove-double-neg N/A

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

   9. /-lowering-/.f64 N/A

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

   10. cbrt-lowering-cbrt.f64 N/A

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

   11. cbrt-lowering-cbrt.f64 N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. *-lowering-*.f64 98.7

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

10. Applied egg-rr 98.7%

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

Alternative 5: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. *-lowering-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

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

   6. metadata-eval 76.4

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

4. Applied egg-rr 76.4%

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

5. Final simplification 76.4%

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

Reproduce

herbie shell --seed 2024197 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))