2-ancestry mixing, zero discriminant

Percentage Accurate: 76.7% → 98.7%

Time: 7.5s

Alternatives: 8

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: 76.7% 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 75.0%

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

   1. associate-/r* N/A

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

   2. cbrt-div N/A

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

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

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

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

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

   5. div-inv N/A

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

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

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

   7. metadata-eval N/A

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

   8. cbrt-lowering-cbrt.f64 98.7

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

4. Applied egg-rr 98.7%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. metadata-eval N/A

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

   4. cbrt-div N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. cbrt-prod N/A

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

   9. div-inv N/A

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

   10. frac-2neg N/A

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

   11. associate-/r/ N/A

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

   12. cbrt-prod N/A

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

   13. pow1/3 N/A

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

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

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

6. Applied egg-rr 98.8%

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

   1. frac-2neg N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. cbrt-prod N/A

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

   5. pow1/3 N/A

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

   6. distribute-frac-neg2 N/A

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

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

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

   8. pow1/3 N/A

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

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

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

   10. distribute-frac-neg2 N/A

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

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

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. /-lowering-/.f64 98.7

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

8. Applied egg-rr 98.7%

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

   1. *-commutative N/A

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

   2. cbrt-unprod N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. clear-num N/A

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

   6. cbrt-div N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

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

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

   12. associate-/r/ N/A

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

   13. metadata-eval N/A

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

   14. cbrt-div N/A

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

   15. frac-2neg N/A

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

   16. cbrt-div N/A

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

   17. clear-num N/A

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

   18. div-inv N/A

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

   19. cbrt-prod N/A

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

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

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

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

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

10. Applied egg-rr 98.8%

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

Alternative 2: 91.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) -2e-304)
   (* (pow (- a) -0.3333333333333333) (cbrt (* g -0.5)))
   (* (pow a -0.3333333333333333) (cbrt (* g 0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -2e-304) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((g * -0.5));
	} else {
		tmp = pow(a, -0.3333333333333333) * cbrt((g * 0.5));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -2e-304) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt((g * -0.5));
	} else {
		tmp = Math.pow(a, -0.3333333333333333) * Math.cbrt((g * 0.5));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= -2e-304)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(Float64(g * -0.5)));
	else
		tmp = Float64((a ^ -0.3333333333333333) * cbrt(Float64(g * 0.5)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], -2e-304], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * -0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -1.99999999999999994e-304

   1. Initial program 69.4%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

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

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

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

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

      5. div-inv N/A

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

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

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

      7. metadata-eval N/A

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

      8. cbrt-lowering-cbrt.f64 98.7

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

   4. Applied egg-rr 98.7%

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

      1. cbrt-undiv N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. pow1/3 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. div-inv N/A

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

      11. distribute-neg-frac2 N/A

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

      12. *-commutative N/A

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

      13. pow-prod-down N/A

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

      14. pow1/3 N/A

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

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

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

      16. distribute-neg-frac2 N/A

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

      17. inv-pow N/A

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

      18. pow-pow N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

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

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

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

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

      23. metadata-eval N/A

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

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

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

   6. Applied egg-rr 92.1%

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

   if -1.99999999999999994e-304 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 80.8%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. *-commutative N/A

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

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

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

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

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

      10. metadata-eval N/A

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

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

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

      12. div-inv N/A

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

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

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

      14. metadata-eval 92.2

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

   4. Applied egg-rr 92.2%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \sqrt{\frac{g \cdot -0.5}{a}}\\ t_2 := \frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\sqrt[3]{-t\_1 \cdot t\_1}\\ \mathbf{elif}\;t\_0 \leq 10^{-312}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+295}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a \cdot 2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0)))
        (t_1 (sqrt (/ (* g -0.5) a)))
        (t_2 (/ (cbrt (* 0.5 (* a (* g a)))) a)))
   (if (<= t_0 (- INFINITY))
     t_2
     (if (<= t_0 -1e-321)
       (cbrt (- (* t_1 t_1)))
       (if (<= t_0 1e-312)
         t_2
         (if (<= t_0 1e+295) (/ 1.0 (cbrt (/ (* a 2.0) g))) t_2))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = sqrt(((g * -0.5) / a));
	double t_2 = cbrt((0.5 * (a * (g * a)))) / a;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_0 <= -1e-321) {
		tmp = cbrt(-(t_1 * t_1));
	} else if (t_0 <= 1e-312) {
		tmp = t_2;
	} else if (t_0 <= 1e+295) {
		tmp = 1.0 / cbrt(((a * 2.0) / g));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = Math.sqrt(((g * -0.5) / a));
	double t_2 = Math.cbrt((0.5 * (a * (g * a)))) / a;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_0 <= -1e-321) {
		tmp = Math.cbrt(-(t_1 * t_1));
	} else if (t_0 <= 1e-312) {
		tmp = t_2;
	} else if (t_0 <= 1e+295) {
		tmp = 1.0 / Math.cbrt(((a * 2.0) / g));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = sqrt(Float64(Float64(g * -0.5) / a))
	t_2 = Float64(cbrt(Float64(0.5 * Float64(a * Float64(g * a)))) / a)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_0 <= -1e-321)
		tmp = cbrt(Float64(-Float64(t_1 * t_1)));
	elseif (t_0 <= 1e-312)
		tmp = t_2;
	elseif (t_0 <= 1e+295)
		tmp = Float64(1.0 / cbrt(Float64(Float64(a * 2.0) / g)));
	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[Sqrt[N[(N[(g * -0.5), $MachinePrecision] / a), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(0.5 * N[(a * N[(g * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$2, If[LessEqual[t$95$0, -1e-321], N[Power[(-N[(t$95$1 * t$95$1), $MachinePrecision]), 1/3], $MachinePrecision], If[LessEqual[t$95$0, 1e-312], t$95$2, If[LessEqual[t$95$0, 1e+295], N[(1.0 / N[Power[N[(N[(a * 2.0), $MachinePrecision] / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -inf.0 or -9.98013e-322 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 9.9999999999847e-313 or 9.9999999999999998e294 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 5.7%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

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

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

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

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

      5. div-inv N/A

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

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

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

      7. metadata-eval N/A

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

      8. cbrt-lowering-cbrt.f64 98.8

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

   4. Applied egg-rr 98.8%

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

   6. Applied egg-rr 66.3%

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

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -9.98013e-322

   1. Initial program 98.4%

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

      1. div-inv N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. pow2 N/A

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

      10. remove-double-neg N/A

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

      11. remove-double-neg N/A

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

      12. sqr-neg N/A

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

      13. pow-prod-down N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. inv-pow N/A

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

   4. Applied egg-rr 98.5%

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

   if 9.9999999999847e-313 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 9.9999999999999998e294

   1. Initial program 98.9%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

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

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

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

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

      5. div-inv N/A

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

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

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

      7. metadata-eval N/A

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

      8. cbrt-lowering-cbrt.f64 98.6

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

   4. Applied egg-rr 98.6%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. div-inv N/A

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

      7. cbrt-div N/A

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

      8. clear-num N/A

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

      9. cbrt-prod N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l/ N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 99.1

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\sqrt[3]{-\sqrt{\frac{g \cdot -0.5}{a}} \cdot \sqrt{\frac{g \cdot -0.5}{a}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 10^{-312}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 10^{+295}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a \cdot 2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\sqrt[3]{t\_0}\\ \mathbf{elif}\;t\_0 \leq 10^{-312}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+295}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a \cdot 2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))) (t_1 (/ (cbrt (* 0.5 (* a (* g a)))) a)))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -1e-321)
       (cbrt t_0)
       (if (<= t_0 1e-312)
         t_1
         (if (<= t_0 1e+295) (/ 1.0 (cbrt (/ (* a 2.0) g))) t_1))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = cbrt((0.5 * (a * (g * a)))) / a;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -1e-321) {
		tmp = cbrt(t_0);
	} else if (t_0 <= 1e-312) {
		tmp = t_1;
	} else if (t_0 <= 1e+295) {
		tmp = 1.0 / cbrt(((a * 2.0) / g));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = Math.cbrt((0.5 * (a * (g * a)))) / a;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= -1e-321) {
		tmp = Math.cbrt(t_0);
	} else if (t_0 <= 1e-312) {
		tmp = t_1;
	} else if (t_0 <= 1e+295) {
		tmp = 1.0 / Math.cbrt(((a * 2.0) / g));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(cbrt(Float64(0.5 * Float64(a * Float64(g * a)))) / a)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -1e-321)
		tmp = cbrt(t_0);
	elseif (t_0 <= 1e-312)
		tmp = t_1;
	elseif (t_0 <= 1e+295)
		tmp = Float64(1.0 / cbrt(Float64(Float64(a * 2.0) / g)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(0.5 * N[(a * N[(g * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -1e-321], N[Power[t$95$0, 1/3], $MachinePrecision], If[LessEqual[t$95$0, 1e-312], t$95$1, If[LessEqual[t$95$0, 1e+295], N[(1.0 / N[Power[N[(N[(a * 2.0), $MachinePrecision] / g), $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 -9.98013e-322 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 9.9999999999847e-313 or 9.9999999999999998e294 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 5.7%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

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

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

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

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

      5. div-inv N/A

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

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

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

      7. metadata-eval N/A

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

      8. cbrt-lowering-cbrt.f64 98.8

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

   4. Applied egg-rr 98.8%

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

   6. Applied egg-rr 66.3%

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

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -9.98013e-322

   1. Initial program 98.4%

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

   if 9.9999999999847e-313 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 9.9999999999999998e294

   1. Initial program 98.9%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

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

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

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

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

      5. div-inv N/A

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

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

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

      7. metadata-eval N/A

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

      8. cbrt-lowering-cbrt.f64 98.6

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

   4. Applied egg-rr 98.6%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. div-inv N/A

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

      7. cbrt-div N/A

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

      8. clear-num N/A

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

      9. cbrt-prod N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l/ N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 99.1

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a \cdot 2}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 10^{-312}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 10^{+295}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a \cdot 2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\sqrt[3]{t\_0}\\ \mathbf{elif}\;t\_0 \leq 10^{-312}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+295}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))) (t_1 (/ (cbrt (* 0.5 (* a (* g a)))) a)))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -1e-321)
       (cbrt t_0)
       (if (<= t_0 1e-312)
         t_1
         (if (<= t_0 1e+295) (/ 1.0 (cbrt (* a (/ 2.0 g)))) t_1))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = cbrt((0.5 * (a * (g * a)))) / a;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -1e-321) {
		tmp = cbrt(t_0);
	} else if (t_0 <= 1e-312) {
		tmp = t_1;
	} else if (t_0 <= 1e+295) {
		tmp = 1.0 / cbrt((a * (2.0 / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = Math.cbrt((0.5 * (a * (g * a)))) / a;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= -1e-321) {
		tmp = Math.cbrt(t_0);
	} else if (t_0 <= 1e-312) {
		tmp = t_1;
	} else if (t_0 <= 1e+295) {
		tmp = 1.0 / Math.cbrt((a * (2.0 / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(cbrt(Float64(0.5 * Float64(a * Float64(g * a)))) / a)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -1e-321)
		tmp = cbrt(t_0);
	elseif (t_0 <= 1e-312)
		tmp = t_1;
	elseif (t_0 <= 1e+295)
		tmp = Float64(1.0 / cbrt(Float64(a * Float64(2.0 / g))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(0.5 * N[(a * N[(g * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -1e-321], N[Power[t$95$0, 1/3], $MachinePrecision], If[LessEqual[t$95$0, 1e-312], t$95$1, If[LessEqual[t$95$0, 1e+295], N[(1.0 / N[Power[N[(a * N[(2.0 / g), $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 -9.98013e-322 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 9.9999999999847e-313 or 9.9999999999999998e294 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 5.7%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

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

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

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

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

      5. div-inv N/A

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

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

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

      7. metadata-eval N/A

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

      8. cbrt-lowering-cbrt.f64 98.8

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

   4. Applied egg-rr 98.8%

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

   6. Applied egg-rr 66.3%

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

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -9.98013e-322

   1. Initial program 98.4%

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

   if 9.9999999999847e-313 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 9.9999999999999998e294

   1. Initial program 98.9%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

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

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

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

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

      5. div-inv N/A

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

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

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

      7. metadata-eval N/A

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

      8. cbrt-lowering-cbrt.f64 98.6

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

   4. Applied egg-rr 98.6%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. div-inv N/A

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

      7. cbrt-div N/A

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

      8. clear-num N/A

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

      9. cbrt-prod N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l/ N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 99.1

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

   6. Applied egg-rr 99.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 99.1

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

   8. Applied egg-rr 99.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a \cdot 2}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 10^{-312}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 10^{+295}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ t_1 := \frac{g}{a \cdot 2}\\ t_2 := \sqrt[3]{t\_1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-312}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+295}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ (cbrt (* 0.5 (* a (* g a)))) a))
        (t_1 (/ g (* a 2.0)))
        (t_2 (cbrt t_1)))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 -1e-321)
       t_2
       (if (<= t_1 1e-312) t_0 (if (<= t_1 1e+295) t_2 t_0))))))

C

double code(double g, double a) {
	double t_0 = cbrt((0.5 * (a * (g * a)))) / a;
	double t_1 = g / (a * 2.0);
	double t_2 = cbrt(t_1);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= -1e-321) {
		tmp = t_2;
	} else if (t_1 <= 1e-312) {
		tmp = t_0;
	} else if (t_1 <= 1e+295) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((0.5 * (a * (g * a)))) / a;
	double t_1 = g / (a * 2.0);
	double t_2 = Math.cbrt(t_1);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_1 <= -1e-321) {
		tmp = t_2;
	} else if (t_1 <= 1e-312) {
		tmp = t_0;
	} else if (t_1 <= 1e+295) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(cbrt(Float64(0.5 * Float64(a * Float64(g * a)))) / a)
	t_1 = Float64(g / Float64(a * 2.0))
	t_2 = cbrt(t_1)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= -1e-321)
		tmp = t_2;
	elseif (t_1 <= 1e-312)
		tmp = t_0;
	elseif (t_1 <= 1e+295)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(N[Power[N[(0.5 * N[(a * N[(g * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$1 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 1/3], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, -1e-321], t$95$2, If[LessEqual[t$95$1, 1e-312], t$95$0, If[LessEqual[t$95$1, 1e+295], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -inf.0 or -9.98013e-322 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 9.9999999999847e-313 or 9.9999999999999998e294 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 5.7%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

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

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

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

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

      5. div-inv N/A

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

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

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

      7. metadata-eval N/A

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

      8. cbrt-lowering-cbrt.f64 98.8

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

   4. Applied egg-rr 98.8%

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

   6. Applied egg-rr 66.3%

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

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -9.98013e-322 or 9.9999999999847e-313 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 9.9999999999999998e294

   1. Initial program 98.6%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a \cdot 2}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 10^{-312}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 10^{+295}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(a \cdot \left(g \cdot a\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Final simplification 75.0%

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

Alternative 8: 76.7% 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 75.0%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. *-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 75.0

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

4. Applied egg-rr 75.0%

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

5. Final simplification 75.0%

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

Reproduce

herbie shell --seed 2024205 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))