2-ancestry mixing, zero discriminant

Percentage Accurate: 75.8% → 98.7%

Time: 6.8s

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: 75.8% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l/ N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

   12. metadata-eval N/A

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

   13. associate-/r/ N/A

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

   14. lift-/.f64 N/A

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

   15. neg-mul-1 N/A

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

   16. lift-neg.f64 N/A

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

   17. div-inv N/A

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

   18. cbrt-prod N/A

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

   19. lower-*.f64 N/A

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

6. Applied rewrites 98.8%

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

Alternative 2: 92.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -5e-308) {
		tmp = cbrt(-g) * pow((a * -2.0), -0.3333333333333333);
	} 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-308) {
		tmp = Math.cbrt(-g) * Math.pow((a * -2.0), -0.3333333333333333);
	} 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-308)
		tmp = Float64(cbrt(Float64(-g)) * (Float64(a * -2.0) ^ -0.3333333333333333));
	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-308], N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[N[(a * -2.0), $MachinePrecision], -0.3333333333333333], $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) < -4.99999999999999955e-308

   1. Initial program 72.4%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cbrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-cbrt.f64 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/r/ N/A

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

      14. lift-/.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lift-neg.f64 N/A

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

      17. div-inv N/A

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

      18. cbrt-prod N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

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

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

      5. metadata-eval N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. metadata-eval N/A

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

      9. remove-double-div N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-div N/A

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

      19. neg-mul-1 N/A

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

      20. associate-/r* N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 N/A

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

      24. metadata-eval 98.8

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

   8. Applied rewrites 98.8%

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

   10. Applied rewrites 92.2%

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

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

   1. Initial program 77.2%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. cbrt-prod N/A

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

      5. pow1/3 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-cbrt.f64 92.6

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

   4. Applied rewrites 92.6%

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

4. Final simplification 92.4%

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

Alternative 3: 92.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -5e-308) {
		tmp = cbrt((g * -0.5)) * pow(-a, -0.3333333333333333);
	} 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-308) {
		tmp = Math.cbrt((g * -0.5)) * Math.pow(-a, -0.3333333333333333);
	} 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-308)
		tmp = Float64(cbrt(Float64(g * -0.5)) * (Float64(-a) ^ -0.3333333333333333));
	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-308], N[(N[Power[N[(g * -0.5), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-a), -0.3333333333333333], $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) < -4.99999999999999955e-308

   1. Initial program 72.4%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cbrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-cbrt.f64 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/r/ N/A

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

      14. lift-/.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lift-neg.f64 N/A

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

      17. div-inv N/A

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

      18. cbrt-prod N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.7

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

      4. lift-cbrt.f64 N/A

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

      5. pow1/3 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. inv-pow N/A

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

      10. pow-pow N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. metadata-eval 92.2

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

   8. Applied rewrites 92.2%

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

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

   1. Initial program 77.2%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. cbrt-prod N/A

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

      5. pow1/3 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-cbrt.f64 92.6

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

   4. Applied rewrites 92.6%

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

4. Final simplification 92.4%

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

Alternative 4: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 1.5 \cdot 10^{-291}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(a \cdot 2\right)}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) 1.5e-291)
   (cbrt (* g (/ 0.5 a)))
   (* (cbrt g) (pow (* a 2.0) -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 1.5e-291) {
		tmp = cbrt((g * (0.5 / a)));
	} 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) <= 1.5e-291) {
		tmp = Math.cbrt((g * (0.5 / a)));
	} 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) <= 1.5e-291)
		tmp = cbrt(Float64(g * Float64(0.5 / a)));
	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], 1.5e-291], N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $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) < 1.5e-291

   1. Initial program 72.8%

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

      1. lift-/.f64 N/A

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

      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 72.8

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

   4. Applied rewrites 72.8%

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

   if 1.5e-291 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 76.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. cbrt-prod N/A

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

      5. pow1/3 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-cbrt.f64 92.6

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

   4. Applied rewrites 92.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 1.5 \cdot 10^{-291}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g} \cdot {\left(a \cdot 2\right)}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 1.5 \cdot 10^{-291}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) 1.5e-291)
   (cbrt (* g (/ 0.5 a)))
   (* (pow a -0.3333333333333333) (cbrt (* g 0.5)))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 1.5e-291) {
		tmp = cbrt((g * (0.5 / a)));
	} 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) <= 1.5e-291) {
		tmp = Math.cbrt((g * (0.5 / a)));
	} 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) <= 1.5e-291)
		tmp = cbrt(Float64(g * Float64(0.5 / a)));
	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], 1.5e-291], N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $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) < 1.5e-291

   1. Initial program 72.8%

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

      1. lift-/.f64 N/A

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

      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 72.8

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

   4. Applied rewrites 72.8%

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

   if 1.5e-291 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 76.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-cbrt.f64 N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 92.6

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

   4. Applied rewrites 92.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 1.5 \cdot 10^{-291}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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(Float64(-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 74.9%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. *-commutative N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. cbrt-div N/A

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

   10. pow1/3 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. clear-num N/A

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

   13. associate-/r/ N/A

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

   14. lift-cbrt.f64 N/A

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

   15. pow1/3 N/A

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

   16. lift-cbrt.f64 N/A

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

   17. cbrt-div N/A

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

   18. lift-/.f64 N/A

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

   19. cbrt-undiv N/A

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

   20. lift-/.f64 N/A

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

   21. frac-2neg N/A

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

   22. lift-neg.f64 N/A

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

   23. associate-/r/ N/A

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

   24. cbrt-prod N/A

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

6. Applied rewrites 98.7%

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

Alternative 7: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Alternative 8: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq 10^{+294}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (/ g (* a 2.0)) 1e+294)
   (cbrt (* g (/ 0.5 a)))
   (/ (cbrt (* g (* 0.5 (* a a)))) a)))

C

double code(double g, double a) {
	double tmp;
	if ((g / (a * 2.0)) <= 1e+294) {
		tmp = cbrt((g * (0.5 / a)));
	} else {
		tmp = cbrt((g * (0.5 * (a * a)))) / a;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((g / (a * 2.0)) <= 1e+294) {
		tmp = Math.cbrt((g * (0.5 / a)));
	} else {
		tmp = Math.cbrt((g * (0.5 * (a * a)))) / a;
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(g / Float64(a * 2.0)) <= 1e+294)
		tmp = cbrt(Float64(g * Float64(0.5 / a)));
	else
		tmp = Float64(cbrt(Float64(g * Float64(0.5 * Float64(a * a)))) / a);
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 1e+294], N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[N[(g * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 1.00000000000000007e294

   1. Initial program 79.6%

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

      1. lift-/.f64 N/A

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

      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 79.6

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

   4. Applied rewrites 79.6%

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

   if 1.00000000000000007e294 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 4.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cbrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-cbrt.f64 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-undiv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/r/ N/A

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

      14. lift-/.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lift-neg.f64 N/A

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

      17. div-inv N/A

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

      18. cbrt-prod N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

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

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

      5. metadata-eval N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. metadata-eval N/A

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

      9. remove-double-div N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-div N/A

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

      19. neg-mul-1 N/A

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

      20. associate-/r* N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 N/A

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

      24. metadata-eval 99.2

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

   8. Applied rewrites 99.2%

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

   10. Applied rewrites 44.6%

       \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq 10^{+294}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. lift-/.f64 N/A

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

   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 74.9

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

4. Applied rewrites 74.9%

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

5. Final simplification 74.9%

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

Reproduce

herbie shell --seed 2024237 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))