2-ancestry mixing, zero discriminant

Percentage Accurate: 76.7% → 98.7%

Time: 7.4s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 76.7% 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

\[\begin{array}{l} \\ \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{g \cdot 0.5} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (* (cbrt (/ 1.0 a)) (cbrt (* g 0.5))))

C

double code(double g, double a) {
	return cbrt((1.0 / a)) * cbrt((g * 0.5));
}

Java

public static double code(double g, double a) {
	return Math.cbrt((1.0 / a)) * Math.cbrt((g * 0.5));
}

Julia

function code(g, a)
	return Float64(cbrt(Float64(1.0 / a)) * cbrt(Float64(g * 0.5)))
end

Wolfram

code[g_, a_] := N[(N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

   1. associate-/r* N/A

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

   2. div-inv N/A

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

   3. cbrt-prod N/A

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

   4. pow1/3 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. inv-pow N/A

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

   8. pow-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-cbrt.f64 N/A

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

   12. div-inv N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 42.5

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

4. Applied egg-rr 42.5%

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

5. Taylor expanded in a around 0

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

   1. lower-cbrt.f64 N/A

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

   2. lower-/.f64 98.8

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

7. Simplified 98.8%

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

Alternative 2: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -5 \cdot 10^{-301}:\\ \;\;\;\;{\left(a \cdot -2\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(a \cdot 2\right)}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) -5e-301)
   (* (pow (* a -2.0) -0.3333333333333333) (cbrt (- g)))
   (* (cbrt g) (pow (* a 2.0) -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -5e-301) {
		tmp = pow((a * -2.0), -0.3333333333333333) * cbrt(-g);
	} else {
		tmp = cbrt(g) * pow((a * 2.0), -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -5e-301) {
		tmp = Math.pow((a * -2.0), -0.3333333333333333) * Math.cbrt(-g);
	} else {
		tmp = Math.cbrt(g) * Math.pow((a * 2.0), -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= -5e-301)
		tmp = Float64((Float64(a * -2.0) ^ -0.3333333333333333) * cbrt(Float64(-g)));
	else
		tmp = Float64(cbrt(g) * (Float64(a * 2.0) ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], -5e-301], N[(N[Power[N[(a * -2.0), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[(-g), 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(a * 2.0), $MachinePrecision], -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -5.00000000000000013e-301

   1. Initial program 73.9%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 0.0

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-cbrt.f64 N/A

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

      2. lower-/.f64 98.8

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

   7. Simplified 98.8%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a}}} \cdot \sqrt[3]{g \cdot 0.5} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cbrt-unprod N/A

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

      4. pow1/3 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. associate-*l/ N/A

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

      18. metadata-eval N/A

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

      19. associate-*r* N/A

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

      20. neg-mul-1 N/A

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

      21. associate-*l/ N/A

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

      22. unpow-prod-down N/A

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

   9. Applied egg-rr 98.8%

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

       1. lift-/.f64 N/A

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

       2. lift-cbrt.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-cbrt.f64 N/A

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

       5. lift-*.f64 98.8

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

       6. lift-cbrt.f64 N/A

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

       7. pow1/3 N/A

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

       8. lift-/.f64 N/A

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

       9. clear-num N/A

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

       10. inv-pow N/A

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

       11. pow-pow N/A

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

       12. lower-pow.f64 N/A

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

       13. div-inv N/A

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

       14. lower-*.f64 N/A

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

       15. metadata-eval N/A

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

       16. metadata-eval 91.9

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

   11. Applied egg-rr 91.9%

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

   if -5.00000000000000013e-301 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 79.7%

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

      1. lift-*.f64 N/A

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

      2. div-inv N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 92.2

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

   4. Applied egg-rr 92.2%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -5 \cdot 10^{-301}:\\ \;\;\;\;{\left(a \cdot -2\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(a \cdot 2\right)}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 2 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(a \cdot 2\right)}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) 2e-299)
   (/ 1.0 (cbrt (/ a (* g 0.5))))
   (* (cbrt g) (pow (* a 2.0) -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 2e-299) {
		tmp = 1.0 / cbrt((a / (g * 0.5)));
	} else {
		tmp = cbrt(g) * pow((a * 2.0), -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 2e-299) {
		tmp = 1.0 / Math.cbrt((a / (g * 0.5)));
	} else {
		tmp = Math.cbrt(g) * Math.pow((a * 2.0), -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= 2e-299)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))));
	else
		tmp = Float64(cbrt(g) * (Float64(a * 2.0) ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], 2e-299], N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(a * 2.0), $MachinePrecision], -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < 1.99999999999999998e-299

   1. Initial program 74.1%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. clear-num N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 74.9

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

   4. Applied egg-rr 74.9%

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

   if 1.99999999999999998e-299 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 79.5%

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

      1. lift-*.f64 N/A

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

      2. div-inv N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 92.3

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

   4. Applied egg-rr 92.3%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 2 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(a \cdot 2\right)}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 2 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) 2e-299)
   (/ 1.0 (cbrt (/ a (* g 0.5))))
   (* (cbrt (* g 0.5)) (pow a -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 2e-299) {
		tmp = 1.0 / cbrt((a / (g * 0.5)));
	} else {
		tmp = cbrt((g * 0.5)) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 2e-299) {
		tmp = 1.0 / Math.cbrt((a / (g * 0.5)));
	} else {
		tmp = Math.cbrt((g * 0.5)) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= 2e-299)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))));
	else
		tmp = Float64(cbrt(Float64(g * 0.5)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], 2e-299], N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < 1.99999999999999998e-299

   1. Initial program 74.1%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. clear-num N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 74.9

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

   4. Applied egg-rr 74.9%

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

   if 1.99999999999999998e-299 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 79.5%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 92.2

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

   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 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 2 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 76.6%

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

   1. associate-/r* N/A

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

   2. div-inv N/A

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

   3. cbrt-prod N/A

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

   4. pow1/3 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. inv-pow N/A

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

   8. pow-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-cbrt.f64 N/A

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

   12. div-inv N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 42.5

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

4. Applied egg-rr 42.5%

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

5. Taylor expanded in a around 0

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

   1. lower-cbrt.f64 N/A

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

   2. lower-/.f64 98.8

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

7. Simplified 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a}}} \cdot \sqrt[3]{g \cdot 0.5} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. cbrt-unprod N/A

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

   4. pow1/3 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/r* N/A

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

   16. metadata-eval N/A

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

   17. associate-*l/ N/A

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

   18. metadata-eval N/A

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

   19. associate-*r* N/A

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

   20. neg-mul-1 N/A

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

   21. associate-*l/ N/A

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

   22. unpow-prod-down N/A

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

9. Applied egg-rr 98.8%

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

    1. lift-/.f64 N/A

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

    2. lift-neg.f64 N/A

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

    3. cbrt-unprod N/A

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

    4. pow1/3 N/A

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

    5. lift-/.f64 N/A

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

    6. frac-2neg N/A

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

    7. metadata-eval N/A

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

    8. associate-/r/ N/A

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

    9. lift-neg.f64 N/A

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

    10. frac-2neg N/A

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

    11. associate-/r/ N/A

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

    12. unpow-prod-down N/A

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

    13. lower-*.f64 N/A

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

11. Applied egg-rr 98.8%

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

Alternative 6: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{g \cdot 0.5}{\sqrt[3]{a \cdot \left(\left(g \cdot 0.5\right) \cdot \left(g \cdot 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{a}}{\frac{2}{g}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))))
   (if (<= t_0 -5e-324)
     (/ 1.0 (cbrt (/ a (* g 0.5))))
     (if (<= t_0 0.0)
       (/ (* g 0.5) (cbrt (* a (* (* g 0.5) (* g 0.5)))))
       (cbrt (/ (/ 1.0 a) (/ 2.0 g)))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double tmp;
	if (t_0 <= -5e-324) {
		tmp = 1.0 / cbrt((a / (g * 0.5)));
	} else if (t_0 <= 0.0) {
		tmp = (g * 0.5) / cbrt((a * ((g * 0.5) * (g * 0.5))));
	} else {
		tmp = cbrt(((1.0 / a) / (2.0 / g)));
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double tmp;
	if (t_0 <= -5e-324) {
		tmp = 1.0 / Math.cbrt((a / (g * 0.5)));
	} else if (t_0 <= 0.0) {
		tmp = (g * 0.5) / Math.cbrt((a * ((g * 0.5) * (g * 0.5))));
	} else {
		tmp = Math.cbrt(((1.0 / a) / (2.0 / g)));
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-324)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(g * 0.5) / cbrt(Float64(a * Float64(Float64(g * 0.5) * Float64(g * 0.5)))));
	else
		tmp = cbrt(Float64(Float64(1.0 / a) / Float64(2.0 / g)));
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-324], N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(g * 0.5), $MachinePrecision] / N[Power[N[(a * N[(N[(g * 0.5), $MachinePrecision] * N[(g * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(1.0 / a), $MachinePrecision] / N[(2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -4.94066e-324

   1. Initial program 84.2%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. clear-num N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 84.4

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

   4. Applied egg-rr 84.4%

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

   if -4.94066e-324 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -0.0

   1. Initial program 4.6%

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

      1. lift-*.f64 N/A

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

      2. cbrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-cbrt.f64 98.7

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

   4. Applied egg-rr 98.7%

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

   6. Applied egg-rr 31.7%

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

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

   1. Initial program 88.3%

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

      1. associate-/r* N/A

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

      2. 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. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.4

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

   4. Applied egg-rr 88.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 0:\\ \;\;\;\;\frac{g \cdot 0.5}{\sqrt[3]{a \cdot \left(\left(g \cdot 0.5\right) \cdot \left(g \cdot 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{a}}{\frac{2}{g}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (/ 1.0 (cbrt (/ a (* g 0.5)))))

C

double code(double g, double a) {
	return 1.0 / cbrt((a / (g * 0.5)));
}

Java

public static double code(double g, double a) {
	return 1.0 / Math.cbrt((a / (g * 0.5)));
}

Julia

function code(g, a)
	return Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))))
end

Wolfram

code[g_, a_] := N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

   1. lift-*.f64 N/A

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

   2. clear-num N/A

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

   3. cbrt-div N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. clear-num N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. div-inv N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 76.7

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

4. Applied egg-rr 76.7%

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

Alternative 8: 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 76.6%

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

3. Final simplification 76.6%

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

Alternative 9: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. lift-*.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 76.6

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

4. Applied egg-rr 76.6%

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

5. Final simplification 76.6%

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

Reproduce

herbie shell --seed 2024210 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))