2-ancestry mixing, zero discriminant

Percentage Accurate: 75.9% → 98.6%

Time: 5.6s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

4. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. cbrt-unprod N/A

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. lift-neg.f64 N/A

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

   10. div-inv N/A

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

   11. frac-2neg N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. lift-neg.f64 N/A

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

   15. cbrt-div N/A

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

   16. lift-neg.f64 N/A

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

   17. remove-double-neg N/A

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

   18. clear-num N/A

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

   19. lower-/.f64 N/A

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

   20. lower-/.f64 N/A

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

   21. lower-cbrt.f64 N/A

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

   22. lower-cbrt.f64 N/A

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

6. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f64 N/A

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

   7. cbrt-div N/A

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

   8. lift-cbrt.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. metadata-eval N/A

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

   11. associate-/r/ N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l/ N/A

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

   14. lift-/.f64 N/A

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

   15. clear-num N/A

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

   16. cbrt-div N/A

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

   17. pow1/3 N/A

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

   18. associate-/r/ N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 N/A

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

   21. lower-cbrt.f64 N/A

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

   22. lift-cbrt.f64 N/A

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

   23. pow1/3 N/A

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

8. Applied rewrites 98.9%

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

9. Final simplification 98.9%

   \[\leadsto \frac{1}{\sqrt[3]{-2 \cdot a} \cdot \sqrt[3]{\frac{-1}{g}}} \]
10. Add Preprocessing

Alternative 2: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq -5 \cdot 10^{-308}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{-0.5 \cdot g}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) -5e-308)
   (* (pow (- a) -0.3333333333333333) (cbrt (* -0.5 g)))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-308) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((-0.5 * g));
	} else {
		tmp = pow((2.0 * a), -0.3333333333333333) * cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= -5e-308) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt((-0.5 * g));
	} else {
		tmp = Math.pow((2.0 * a), -0.3333333333333333) * Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= -5e-308)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(Float64(-0.5 * g)));
	else
		tmp = Float64((Float64(2.0 * a) ^ -0.3333333333333333) * cbrt(g));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], -5e-308], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[N[(-0.5 * g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -4.99999999999999955e-308

   1. Initial program 81.2%

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

   4. Applied rewrites 98.7%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-neg.f64 N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. pow1/3 N/A

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

      9. pow-flip N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval 92.5

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

   6. Applied rewrites 92.5%

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

   if -4.99999999999999955e-308 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 77.1%

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

   4. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. cbrt-prod N/A

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

      5. pow1/3 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. pow1/3 N/A

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

      10. cbrt-unprod N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. metadata-eval N/A

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

      14. lift-neg.f64 N/A

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

      15. associate-*l/ N/A

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

      16. metadata-eval N/A

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

      17. lower-cbrt.f64 N/A

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

      18. frac-2neg N/A

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

      19. lift-neg.f64 N/A

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

      20. remove-double-neg N/A

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

      21. lower-/.f64 N/A

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

      22. metadata-eval N/A

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

      23. lower-cbrt.f64 98.8

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

   6. Applied rewrites 98.8%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 92.1

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

   8. Applied rewrites 92.1%

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

Alternative 3: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 5 \cdot 10^{-264}:\\ \;\;\;\;\sqrt[3]{\frac{g}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* 2.0 a) 5e-264)
   (cbrt (/ g (* 2.0 a)))
   (* (pow (* 2.0 a) -0.3333333333333333) (cbrt g))))

C

double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= 5e-264) {
		tmp = cbrt((g / (2.0 * a)));
	} else {
		tmp = pow((2.0 * a), -0.3333333333333333) * cbrt(g);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((2.0 * a) <= 5e-264) {
		tmp = Math.cbrt((g / (2.0 * a)));
	} else {
		tmp = Math.pow((2.0 * a), -0.3333333333333333) * Math.cbrt(g);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= 5e-264)
		tmp = cbrt(Float64(g / Float64(2.0 * a)));
	else
		tmp = Float64((Float64(2.0 * a) ^ -0.3333333333333333) * cbrt(g));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], 5e-264], N[Power[N[(g / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[N[(2.0 * a), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < 5.0000000000000001e-264

   1. Initial program 81.8%

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

   if 5.0000000000000001e-264 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 76.2%

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

   4. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. cbrt-prod N/A

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

      5. pow1/3 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. pow1/3 N/A

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

      10. cbrt-unprod N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. metadata-eval N/A

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

      14. lift-neg.f64 N/A

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

      15. associate-*l/ N/A

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

      16. metadata-eval N/A

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

      17. lower-cbrt.f64 N/A

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

      18. frac-2neg N/A

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

      19. lift-neg.f64 N/A

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

      20. remove-double-neg N/A

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

      21. lower-/.f64 N/A

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

      22. metadata-eval N/A

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

      23. lower-cbrt.f64 98.8

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

   6. Applied rewrites 98.8%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 92.2

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

   8. Applied rewrites 92.2%

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

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

   \[\leadsto \sqrt[3]{-0.5 \cdot g} \cdot \sqrt[3]{\frac{-1}{a}} \]
6. Add Preprocessing

Alternative 5: 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 79.3%

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

4. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. cbrt-prod N/A

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

   5. pow1/3 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. pow1/3 N/A

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

   10. cbrt-unprod N/A

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

   11. lift-/.f64 N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-neg.f64 N/A

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

   15. associate-*l/ N/A

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

   16. metadata-eval N/A

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

   17. lower-cbrt.f64 N/A

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

   18. frac-2neg N/A

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

   19. lift-neg.f64 N/A

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

   20. remove-double-neg N/A

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

   21. lower-/.f64 N/A

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

   22. metadata-eval N/A

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

   23. lower-cbrt.f64 98.8

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

6. Applied rewrites 98.8%

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

7. Final simplification 98.8%

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

Alternative 6: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

Alternative 7: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

4. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-unprod N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. cbrt-prod N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. associate-/r/ N/A

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

   12. clear-num N/A

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

   13. lift-/.f64 N/A

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

   14. cbrt-prod N/A

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

   15. neg-mul-1 N/A

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

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

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

   17. lift-neg.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. lift-cbrt.f64 79.3

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

   20. lift-*.f64 N/A

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

   21. lift-neg.f64 N/A

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

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

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

   23. lift-/.f64 N/A

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

6. Applied rewrites 79.3%

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

Reproduce

herbie shell --seed 2024267 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))