2-ancestry mixing, zero discriminant

Percentage Accurate: 76.9% → 98.7%

Time: 8.7s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{\frac{1}{\frac{2}{g}}}}{\sqrt[3]{a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ (cbrt (/ 1.0 (/ 2.0 g))) (cbrt a)))

C

double code(double g, double a) {
	return cbrt((1.0 / (2.0 / g))) / cbrt(a);
}

Java

public static double code(double g, double a) {
	return Math.cbrt((1.0 / (2.0 / g))) / Math.cbrt(a);
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(1.0 / Float64(2.0 / g))) / cbrt(a))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(1.0 / N[(2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

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

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. cbrt-div N/A

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

   5. associate-*l* N/A

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

   6. cbrt-div N/A

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

   7. metadata-eval N/A

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

   8. pow1/3 N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

   11. pow-prod-down N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. pow-pow N/A

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

   17. inv-pow N/A

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

   18. clear-num N/A

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

   19. pow1/3 N/A

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

   20. cbrt-div N/A

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

   21. unpow1/3 N/A

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

   22. associate-*r/ N/A

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

6. Applied egg-rr 98.7%

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

   1. +-rgt-identity N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. --rgt-identity N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. --rgt-identity N/A

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

   8. pow2 N/A

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

   9. pow-div N/A

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

   10. metadata-eval N/A

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

   11. inv-pow N/A

       \[\leadsto \frac{\sqrt[3]{\frac{1}{\color{blue}{\frac{1}{g \cdot \frac{1}{2} + 0}}}}}{\sqrt[3]{a}} \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{1}{\frac{1}{g \cdot \frac{1}{2} + 0}}}}}{\sqrt[3]{a}} \]

   13. +-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

   16. metadata-eval N/A

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

   17. /-lowering-/.f64 98.8

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

8. Applied egg-rr 98.8%

   \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{1}{\frac{2}{g}}}}}{\sqrt[3]{a}} \]
9. Add Preprocessing

Alternative 2: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{2 \cdot a}\\ t_1 := \frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{0 - \mathsf{fma}\left(g \cdot a, \mathsf{fma}\left(g, -0.25, 0\right), 0\right)}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{elif}\;t\_0 \leq 10^{-319}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{a}}{\frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* 2.0 a)))
        (t_1
         (/
          (fma g 0.5 0.0)
          (cbrt (- 0.0 (fma (* g a) (fma g -0.25 0.0) 0.0))))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -2e-300)
       (cbrt (* g (/ 0.5 a)))
       (if (<= t_0 1e-319)
         t_1
         (if (<= t_0 5e+284) (cbrt (/ (/ 1.0 a) (/ 2.0 g))) t_1))))))

C

double code(double g, double a) {
	double t_0 = g / (2.0 * a);
	double t_1 = fma(g, 0.5, 0.0) / cbrt((0.0 - fma((g * a), fma(g, -0.25, 0.0), 0.0)));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -2e-300) {
		tmp = cbrt((g * (0.5 / a)));
	} else if (t_0 <= 1e-319) {
		tmp = t_1;
	} else if (t_0 <= 5e+284) {
		tmp = cbrt(((1.0 / a) / (2.0 / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(2.0 * a))
	t_1 = Float64(fma(g, 0.5, 0.0) / cbrt(Float64(0.0 - fma(Float64(g * a), fma(g, -0.25, 0.0), 0.0))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -2e-300)
		tmp = cbrt(Float64(g * Float64(0.5 / a)));
	elseif (t_0 <= 1e-319)
		tmp = t_1;
	elseif (t_0 <= 5e+284)
		tmp = cbrt(Float64(Float64(1.0 / a) / Float64(2.0 / g)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(g * 0.5 + 0.0), $MachinePrecision] / N[Power[N[(0.0 - N[(N[(g * a), $MachinePrecision] * N[(g * -0.25 + 0.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -2e-300], N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t$95$0, 1e-319], t$95$1, If[LessEqual[t$95$0, 5e+284], N[Power[N[(N[(1.0 / a), $MachinePrecision] / N[(2.0 / g), $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)) < -inf.0 or -2.00000000000000005e-300 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 9.99989e-320 or 4.9999999999999999e284 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 11.1%

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

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. cbrt-div N/A

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

      5. associate-*l* N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. pow1/3 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-down N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. pow-pow N/A

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

      17. inv-pow N/A

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

      18. clear-num N/A

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

      19. pow1/3 N/A

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

      20. cbrt-div N/A

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

      21. unpow1/3 N/A

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

      22. associate-*r/ N/A

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

   6. Applied egg-rr 98.7%

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

      1. +-rgt-identity N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. --rgt-identity N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. --rgt-identity N/A

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

      8. pow2 N/A

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

      9. pow-div N/A

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

      10. metadata-eval N/A

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

      11. inv-pow N/A

          \[\leadsto \frac{\sqrt[3]{\frac{1}{\color{blue}{\frac{1}{g \cdot \frac{1}{2} + 0}}}}}{\sqrt[3]{a}} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{1}{\frac{1}{g \cdot \frac{1}{2} + 0}}}}}{\sqrt[3]{a}} \]

      13. +-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. /-lowering-/.f64 98.9

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

   8. Applied egg-rr 98.9%

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

   10. Applied egg-rr 62.7%

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

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -2.00000000000000005e-300

   1. Initial program 99.1%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 99.1

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

   4. Applied egg-rr 99.1%

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

   if 9.99989e-320 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 4.9999999999999999e284

   1. Initial program 99.1%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/r* N/A

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

      4. associate-/l/ N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 99.1

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

   4. Applied egg-rr 99.1%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{a}}{\frac{2}{g}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{2 \cdot a} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{0 - \mathsf{fma}\left(g \cdot a, \mathsf{fma}\left(g, -0.25, 0\right), 0\right)}}\\ \mathbf{elif}\;\frac{g}{2 \cdot a} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{elif}\;\frac{g}{2 \cdot a} \leq 10^{-319}:\\ \;\;\;\;\frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{0 - \mathsf{fma}\left(g \cdot a, \mathsf{fma}\left(g, -0.25, 0\right), 0\right)}}\\ \mathbf{elif}\;\frac{g}{2 \cdot a} \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{a}}{\frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(g, 0.5, 0\right)}{\sqrt[3]{0 - \mathsf{fma}\left(g \cdot a, \mathsf{fma}\left(g, -0.25, 0\right), 0\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 78.1%

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

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. cbrt-div N/A

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

   5. associate-*l* N/A

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

   6. cbrt-div N/A

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

   7. metadata-eval N/A

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

   8. pow1/3 N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

   11. pow-prod-down N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. pow-pow N/A

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

   17. inv-pow N/A

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

   18. clear-num N/A

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

   19. pow1/3 N/A

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

   20. cbrt-div N/A

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

   21. unpow1/3 N/A

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

   22. associate-*r/ N/A

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

6. Applied egg-rr 98.7%

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

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 98.7

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

8. Applied egg-rr 98.7%

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

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. cbrt-div N/A

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

   5. associate-*l* N/A

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

   6. cbrt-div N/A

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

   7. metadata-eval N/A

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

   8. pow1/3 N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

   11. pow-prod-down N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. pow-pow N/A

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

   17. inv-pow N/A

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

   18. clear-num N/A

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

   19. pow1/3 N/A

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

   20. cbrt-div N/A

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

   21. unpow1/3 N/A

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

   22. associate-*r/ N/A

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

6. Applied egg-rr 98.7%

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

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. cbrt-div N/A

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

   8. associate-*l/ N/A

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

   9. associate-/r/ N/A

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. cbrt-prod N/A

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

   13. metadata-eval N/A

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

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

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

   15. *-rgt-identity N/A

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

   16. pow1/3 N/A

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

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

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

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

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

   19. distribute-neg-frac N/A

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

   20. metadata-eval N/A

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

   21. /-lowering-/.f64 N/A

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

   22. pow1/3 N/A

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

   23. cbrt-lowering-cbrt.f64 98.7

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

8. Applied egg-rr 98.7%

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

Alternative 5: 76.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{\frac{1}{a}}{\frac{2}{g}}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ (/ 1.0 a) (/ 2.0 g))))

C

double code(double g, double a) {
	return cbrt(((1.0 / a) / (2.0 / g)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt(((1.0 / a) / (2.0 / g)));
}

Julia

function code(g, a)
	return cbrt(Float64(Float64(1.0 / a) / Float64(2.0 / g)))
end

Wolfram

code[g_, a_] := N[Power[N[(N[(1.0 / a), $MachinePrecision] / N[(2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 78.1%

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

   1. associate-/r* N/A

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

   2. clear-num N/A

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

   3. associate-/r* N/A

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

   4. associate-/l/ N/A

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

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

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

   6. /-lowering-/.f64 N/A

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

   7. /-lowering-/.f64 78.5

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

4. Applied egg-rr 78.5%

   \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{a}}{\frac{2}{g}}}} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. associate-/l/ N/A

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

   2. div-inv N/A

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

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

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

   4. /-lowering-/.f64 N/A

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

   5. metadata-eval 78.5

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

4. Applied egg-rr 78.5%

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

5. Final simplification 78.5%

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

Alternative 7: 76.9% 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 78.1%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 78.5

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

4. Applied egg-rr 78.5%

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

5. Final simplification 78.5%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))