2-ancestry mixing, zero discriminant

Percentage Accurate: 76.0% → 98.7%

Time: 8.6s

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: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, a)
	return Float64(cbrt(Float64(-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 79.5%

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

   1. associate-/r* N/A

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

   2. cbrt-div N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. div-inv N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

6. Applied rewrites 98.8%

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

Alternative 2: 89.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a} \cdot \left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))) (t_1 (/ (cbrt (* a (* a (* g 0.5)))) a)))
   (if (<= t_0 -1e+255)
     t_1
     (if (<= t_0 -2e-302)
       (/ 1.0 (cbrt (/ a (* g 0.5))))
       (if (<= t_0 0.0)
         t_1
         (if (<= t_0 2e+277)
           (cbrt (* (/ g a) (* (sqrt 0.5) (sqrt 0.5))))
           t_1))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = cbrt((a * (a * (g * 0.5)))) / a;
	double tmp;
	if (t_0 <= -1e+255) {
		tmp = t_1;
	} else if (t_0 <= -2e-302) {
		tmp = 1.0 / cbrt((a / (g * 0.5)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+277) {
		tmp = cbrt(((g / a) * (sqrt(0.5) * sqrt(0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = Math.cbrt((a * (a * (g * 0.5)))) / a;
	double tmp;
	if (t_0 <= -1e+255) {
		tmp = t_1;
	} else if (t_0 <= -2e-302) {
		tmp = 1.0 / Math.cbrt((a / (g * 0.5)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+277) {
		tmp = Math.cbrt(((g / a) * (Math.sqrt(0.5) * Math.sqrt(0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(cbrt(Float64(a * Float64(a * Float64(g * 0.5)))) / a)
	tmp = 0.0
	if (t_0 <= -1e+255)
		tmp = t_1;
	elseif (t_0 <= -2e-302)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+277)
		tmp = cbrt(Float64(Float64(g / a) * Float64(sqrt(0.5) * sqrt(0.5))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(a * N[(a * N[(g * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+255], t$95$1, If[LessEqual[t$95$0, -2e-302], N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+277], N[Power[N[(N[(g / a), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -9.99999999999999988e254 or -1.9999999999999999e-302 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 0.0 or 2.00000000000000001e277 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 14.3%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-cbrt.f64 98.6

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

   4. Applied rewrites 98.6%

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

   6. Applied rewrites 98.6%

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

   8. Applied rewrites 83.3%

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

   if -9.99999999999999988e254 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -1.9999999999999999e-302

   1. Initial program 99.0%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 99.2

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

   4. Applied rewrites 99.2%

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

   if 0.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 2.00000000000000001e277

   1. Initial program 98.1%

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

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 97.1

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

   4. Applied rewrites 97.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. sqrt-pow1 N/A

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

      6. inv-pow N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. sqrt-prod N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. *-commutative N/A

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

      16. sqrt-pow1 N/A

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

      17. inv-pow N/A

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

      18. clear-num N/A

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

      19. div-inv N/A

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

      20. lift-/.f64 N/A

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

      21. clear-num N/A

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

      22. lift-/.f64 N/A

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

      23. sqrt-prod N/A

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

      24. lift-sqrt.f64 N/A

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

      25. lift-sqrt.f64 N/A

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

   6. Applied rewrites 98.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -1 \cdot 10^{+255}:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a} \cdot \left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-294}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) -4e-294)
   (* (pow (- a) -0.3333333333333333) (cbrt (* g -0.5)))
   (* (cbrt (* g 0.5)) (pow a -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -4e-294) {
		tmp = pow(-a, -0.3333333333333333) * cbrt((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) <= -4e-294) {
		tmp = Math.pow(-a, -0.3333333333333333) * Math.cbrt((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) <= -4e-294)
		tmp = Float64((Float64(-a) ^ -0.3333333333333333) * cbrt(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], -4e-294], N[(N[Power[(-a), -0.3333333333333333], $MachinePrecision] * N[Power[N[(g * -0.5), $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) < -4.00000000000000007e-294

   1. Initial program 78.2%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-cbrt.f64 98.8

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

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 98.9%

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

      1. frac-2neg N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

      4. cbrt-prod N/A

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

      5. pow1/3 N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/3 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. inv-pow N/A

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

      11. metadata-eval N/A

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

      12. pow-div N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. cube-unmult N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-frac-neg N/A

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

      19. lower-cbrt.f64 N/A

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

      20. distribute-frac-neg N/A

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

      21. lift-*.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. cube-unmult N/A

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

      26. pow-div N/A

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

   8. Applied rewrites 98.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cbrt.f64 N/A

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

      10. pow1/3 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. metadata-eval N/A

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

      14. inv-pow N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. metadata-eval N/A

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

      21. lift-cbrt.f64 N/A

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

      22. lift-cbrt.f64 N/A

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

      23. cbrt-unprod N/A

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

      24. lift-neg.f64 N/A

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

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

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

      26. lift-*.f64 N/A

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

   10. Applied rewrites 92.3%

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

   if -4.00000000000000007e-294 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 80.7%

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

      1. associate-/r* N/A

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

      2. 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 92.4

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

   4. Applied rewrites 92.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq -4 \cdot 10^{-294}:\\ \;\;\;\;{\left(-a\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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. cbrt-div N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. div-inv N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-cbrt.f64 98.7

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

4. Applied rewrites 98.7%

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

Alternative 5: 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 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. 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 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 6: 89.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\sqrt[3]{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))) (t_1 (/ (cbrt (* a (* a (* g 0.5)))) a)))
   (if (<= t_0 -1e+255)
     t_1
     (if (<= t_0 -2e-302)
       (/ 1.0 (cbrt (/ a (* g 0.5))))
       (if (<= t_0 0.0) t_1 (if (<= t_0 2e+277) (cbrt t_0) t_1))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = cbrt((a * (a * (g * 0.5)))) / a;
	double tmp;
	if (t_0 <= -1e+255) {
		tmp = t_1;
	} else if (t_0 <= -2e-302) {
		tmp = 1.0 / cbrt((a / (g * 0.5)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+277) {
		tmp = cbrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = Math.cbrt((a * (a * (g * 0.5)))) / a;
	double tmp;
	if (t_0 <= -1e+255) {
		tmp = t_1;
	} else if (t_0 <= -2e-302) {
		tmp = 1.0 / Math.cbrt((a / (g * 0.5)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+277) {
		tmp = Math.cbrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(cbrt(Float64(a * Float64(a * Float64(g * 0.5)))) / a)
	tmp = 0.0
	if (t_0 <= -1e+255)
		tmp = t_1;
	elseif (t_0 <= -2e-302)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+277)
		tmp = cbrt(t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(a * N[(a * N[(g * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+255], t$95$1, If[LessEqual[t$95$0, -2e-302], N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+277], N[Power[t$95$0, 1/3], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -9.99999999999999988e254 or -1.9999999999999999e-302 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 0.0 or 2.00000000000000001e277 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 14.3%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-cbrt.f64 98.6

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

   4. Applied rewrites 98.6%

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

   6. Applied rewrites 98.6%

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

   8. Applied rewrites 83.3%

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

   if -9.99999999999999988e254 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -1.9999999999999999e-302

   1. Initial program 99.0%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 99.2

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

   4. Applied rewrites 99.2%

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

   if 0.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 2.00000000000000001e277

   1. Initial program 98.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -1 \cdot 10^{+255}:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{\frac{a}{g}}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\sqrt[3]{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))) (t_1 (/ (cbrt (* a (* a (* g 0.5)))) a)))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -2e-302)
       (cbrt (/ 0.5 (/ a g)))
       (if (<= t_0 0.0) t_1 (if (<= t_0 2e+277) (cbrt t_0) t_1))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = cbrt((a * (a * (g * 0.5)))) / a;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -2e-302) {
		tmp = cbrt((0.5 / (a / g)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+277) {
		tmp = cbrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = Math.cbrt((a * (a * (g * 0.5)))) / a;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= -2e-302) {
		tmp = Math.cbrt((0.5 / (a / g)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+277) {
		tmp = Math.cbrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(cbrt(Float64(a * Float64(a * Float64(g * 0.5)))) / a)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -2e-302)
		tmp = cbrt(Float64(0.5 / Float64(a / g)));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+277)
		tmp = cbrt(t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(a * N[(a * N[(g * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -2e-302], N[Power[N[(0.5 / N[(a / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+277], N[Power[t$95$0, 1/3], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -inf.0 or -1.9999999999999999e-302 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 0.0 or 2.00000000000000001e277 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 8.1%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-cbrt.f64 98.5

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

   4. Applied rewrites 98.5%

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

   6. Applied rewrites 98.5%

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

   8. Applied rewrites 82.1%

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

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -1.9999999999999999e-302

   1. Initial program 99.0%

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

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

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

   if 0.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 2.00000000000000001e277

   1. Initial program 98.1%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{\frac{a}{g}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(a \cdot \left(g \cdot 0.5\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.0% 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 79.5%

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

3. Final simplification 79.5%

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

Alternative 9: 76.0% 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 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. 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 79.5

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

4. Applied rewrites 79.5%

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

5. Final simplification 79.5%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))