2-ancestry mixing, zero discriminant

Percentage Accurate: 76.1% → 98.7%

Time: 6.9s

Alternatives: 11

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: 76.1% 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

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

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 78.2%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. 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. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. div-inv N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-cbrt.f64 98.8

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

4. Applied rewrites 98.8%

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

Alternative 2: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{1}{\frac{a}{\sqrt[3]{a \cdot \left(g \cdot \left(0.5 \cdot a\right)\right)}}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{g}{a}}{\frac{0.5}{a} \cdot \left(a \cdot 4\right)}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-310}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))) (t_1 (/ 1.0 (/ a (cbrt (* a (* g (* 0.5 a))))))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -5e-306)
       (cbrt (/ (/ g a) (* (/ 0.5 a) (* a 4.0))))
       (if (<= t_0 2e-310)
         t_1
         (if (<= t_0 2e+302) (/ 1.0 (cbrt (* a (/ 2.0 g)))) t_1))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = 1.0 / (a / cbrt((a * (g * (0.5 * a)))));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -5e-306) {
		tmp = cbrt(((g / a) / ((0.5 / a) * (a * 4.0))));
	} else if (t_0 <= 2e-310) {
		tmp = t_1;
	} else if (t_0 <= 2e+302) {
		tmp = 1.0 / cbrt((a * (2.0 / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = 1.0 / (a / Math.cbrt((a * (g * (0.5 * a)))));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= -5e-306) {
		tmp = Math.cbrt(((g / a) / ((0.5 / a) * (a * 4.0))));
	} else if (t_0 <= 2e-310) {
		tmp = t_1;
	} else if (t_0 <= 2e+302) {
		tmp = 1.0 / Math.cbrt((a * (2.0 / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(1.0 / Float64(a / cbrt(Float64(a * Float64(g * Float64(0.5 * a))))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -5e-306)
		tmp = cbrt(Float64(Float64(g / a) / Float64(Float64(0.5 / a) * Float64(a * 4.0))));
	elseif (t_0 <= 2e-310)
		tmp = t_1;
	elseif (t_0 <= 2e+302)
		tmp = Float64(1.0 / cbrt(Float64(a * Float64(2.0 / g))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(a / N[Power[N[(a * N[(g * N[(0.5 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -5e-306], N[Power[N[(N[(g / a), $MachinePrecision] / N[(N[(0.5 / a), $MachinePrecision] * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t$95$0, 2e-310], t$95$1, If[LessEqual[t$95$0, 2e+302], N[(1.0 / N[Power[N[(a * N[(2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -inf.0 or -4.99999999999999998e-306 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 1.999999999999994e-310 or 2.0000000000000002e302 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 8.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. 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 98.6%

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

   8. Applied rewrites 32.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 77.4

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

   10. Applied rewrites 77.4%

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

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -4.99999999999999998e-306

   1. Initial program 99.1%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto \sqrt[3]{\frac{g \cdot \left(\color{blue}{\frac{2}{4}} \cdot 1\right)}{a}} \]

      7. *-inverses N/A

         \[\leadsto \sqrt[3]{\frac{g \cdot \left(\frac{2}{4} \cdot \color{blue}{\frac{a}{a}}\right)}{a}} \]

      8. times-frac N/A

         \[\leadsto \sqrt[3]{\frac{g \cdot \color{blue}{\frac{2 \cdot a}{4 \cdot a}}}{a}} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{g \cdot \frac{\color{blue}{2 \cdot a}}{4 \cdot a}}{a}} \]

      10. *-commutative N/A

          \[\leadsto \sqrt[3]{\frac{g \cdot \frac{2 \cdot a}{\color{blue}{a \cdot 4}}}{a}} \]

      11. associate-*l/ N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a} \cdot \frac{2 \cdot a}{a \cdot 4}}} \]

      12. times-frac N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{g \cdot \left(2 \cdot a\right)}{a \cdot \left(a \cdot 4\right)}}} \]

      13. associate-/r* N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g \cdot \left(2 \cdot a\right)}{a}}{a \cdot 4}}} \]

      14. *-commutative N/A

          \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{\left(2 \cdot a\right) \cdot g}}{a}}{a \cdot 4}} \]

      15. associate-/l* N/A

          \[\leadsto \sqrt[3]{\frac{\color{blue}{\left(2 \cdot a\right) \cdot \frac{g}{a}}}{a \cdot 4}} \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{\left(2 \cdot a\right) \cdot \color{blue}{\frac{g}{a}}}{a \cdot 4}} \]

      17. associate-*r/ N/A

          \[\leadsto \sqrt[3]{\color{blue}{\left(2 \cdot a\right) \cdot \frac{\frac{g}{a}}{a \cdot 4}}} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt[3]{\color{blue}{\left(2 \cdot a\right)} \cdot \frac{\frac{g}{a}}{a \cdot 4}} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt[3]{\left(\color{blue}{\frac{1}{\frac{1}{2}}} \cdot a\right) \cdot \frac{\frac{g}{a}}{a \cdot 4}} \]

      20. associate-/r/ N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\frac{\frac{1}{2}}{a}}} \cdot \frac{\frac{g}{a}}{a \cdot 4}} \]

      21. lift-/.f64 N/A

          \[\leadsto \sqrt[3]{\frac{1}{\color{blue}{\frac{\frac{1}{2}}{a}}} \cdot \frac{\frac{g}{a}}{a \cdot 4}} \]

      22. frac-times N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{1 \cdot \frac{g}{a}}{\frac{\frac{1}{2}}{a} \cdot \left(a \cdot 4\right)}}} \]

      23. *-lft-identity N/A

          \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{g}{a}}}{\frac{\frac{1}{2}}{a} \cdot \left(a \cdot 4\right)}} \]

      24. lower-/.f64 N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{a}}{\frac{\frac{1}{2}}{a} \cdot \left(a \cdot 4\right)}}} \]

      25. lower-*.f64 N/A

          \[\leadsto \sqrt[3]{\frac{\frac{g}{a}}{\color{blue}{\frac{\frac{1}{2}}{a} \cdot \left(a \cdot 4\right)}}} \]

      26. lower-*.f64 99.1

          \[\leadsto \sqrt[3]{\frac{\frac{g}{a}}{\frac{0.5}{a} \cdot \color{blue}{\left(a \cdot 4\right)}}} \]

   6. Applied rewrites 99.1%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{a}}{\frac{0.5}{a} \cdot \left(a \cdot 4\right)}}} \]

   if 1.999999999999994e-310 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 2.0000000000000002e302

   1. Initial program 98.8%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. 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 98.8%

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

   8. Applied rewrites 51.3%

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

   10. Applied rewrites 99.1%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt[3]{a \cdot \left(g \cdot \left(0.5 \cdot a\right)\right)}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{g}{a}}{\frac{0.5}{a} \cdot \left(a \cdot 4\right)}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt[3]{a \cdot \left(g \cdot \left(0.5 \cdot a\right)\right)}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt[3]{a \cdot \left(g \cdot \left(0.5 \cdot a\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{1}{\frac{a}{\sqrt[3]{a \cdot \left(g \cdot \left(0.5 \cdot a\right)\right)}}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\sqrt[3]{t\_0}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-310}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0))) (t_1 (/ 1.0 (/ a (cbrt (* a (* g (* 0.5 a))))))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -5e-306)
       (cbrt t_0)
       (if (<= t_0 2e-310)
         t_1
         (if (<= t_0 2e+302) (/ 1.0 (cbrt (* a (/ 2.0 g)))) t_1))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = 1.0 / (a / cbrt((a * (g * (0.5 * a)))));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -5e-306) {
		tmp = cbrt(t_0);
	} else if (t_0 <= 2e-310) {
		tmp = t_1;
	} else if (t_0 <= 2e+302) {
		tmp = 1.0 / cbrt((a * (2.0 / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = 1.0 / (a / Math.cbrt((a * (g * (0.5 * a)))));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= -5e-306) {
		tmp = Math.cbrt(t_0);
	} else if (t_0 <= 2e-310) {
		tmp = t_1;
	} else if (t_0 <= 2e+302) {
		tmp = 1.0 / Math.cbrt((a * (2.0 / g)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(1.0 / Float64(a / cbrt(Float64(a * Float64(g * Float64(0.5 * a))))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -5e-306)
		tmp = cbrt(t_0);
	elseif (t_0 <= 2e-310)
		tmp = t_1;
	elseif (t_0 <= 2e+302)
		tmp = Float64(1.0 / cbrt(Float64(a * Float64(2.0 / g))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(a / N[Power[N[(a * N[(g * N[(0.5 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -5e-306], N[Power[t$95$0, 1/3], $MachinePrecision], If[LessEqual[t$95$0, 2e-310], t$95$1, If[LessEqual[t$95$0, 2e+302], N[(1.0 / N[Power[N[(a * N[(2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -inf.0 or -4.99999999999999998e-306 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 1.999999999999994e-310 or 2.0000000000000002e302 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 8.6%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. 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 98.6%

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

   8. Applied rewrites 32.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 77.4

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

   10. Applied rewrites 77.4%

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

   if -inf.0 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -4.99999999999999998e-306

   1. Initial program 99.1%

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

   if 1.999999999999994e-310 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 2.0000000000000002e302

   1. Initial program 98.8%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. 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 98.8%

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

   8. Applied rewrites 51.3%

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

   10. Applied rewrites 99.1%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt[3]{a \cdot \left(g \cdot \left(0.5 \cdot a\right)\right)}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\sqrt[3]{\frac{g}{a \cdot 2}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt[3]{a \cdot \left(g \cdot \left(0.5 \cdot a\right)\right)}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt[3]{a \cdot \left(g \cdot \left(0.5 \cdot a\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) -1e-299)
   (* (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) <= -1e-299) {
		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) <= -1e-299) {
		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) <= -1e-299)
		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], -1e-299], 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) < -9.99999999999999992e-300

   1. Initial program 82.3%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 98.9

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

   4. Applied rewrites 98.9%

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

   6. Applied rewrites 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 92.2

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

   8. Applied rewrites 92.2%

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

   if -9.99999999999999992e-300 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 73.9%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 92.2

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

   4. Applied rewrites 92.2%

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

4. Final simplification 92.2%

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

Alternative 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 78.2%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. 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 6: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.2%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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. 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. cbrt-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. div-inv N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-cbrt.f64 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l/ N/A

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

   8. lift-/.f64 N/A

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

   9. cbrt-prod N/A

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

   10. pow1/3 N/A

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

   11. pow1/3 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. metadata-eval N/A

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

   15. associate-/r* N/A

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

   16. lift-*.f64 N/A

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

   17. inv-pow N/A

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

   18. pow-pow N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-pow.f64 N/A

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 N/A

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

   25. metadata-eval N/A

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

   26. pow1/3 N/A

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

   27. lower-cbrt.f64 45.4

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

6. Applied rewrites 45.4%

   \[\leadsto \color{blue}{{\left(a \cdot 2\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. inv-pow N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. *-inverses N/A

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

   12. times-frac N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/l* N/A

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

   15. *-lft-identity N/A

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

   16. *-inverses N/A

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

   17. times-frac N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. associate-/l* N/A

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

   21. lift-*.f64 N/A

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

   22. lift-/.f64 N/A

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

   23. pow1/3 N/A

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

   24. lower-cbrt.f64 36.4

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

8. Applied rewrites 98.7%

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

9. Final simplification 98.7%

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

Alternative 7: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{g \cdot \left(a \cdot \left(0.5 \cdot a\right)\right)}\\ t_1 := \frac{g}{a \cdot 2}\\ t_2 := \frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-306}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-290}:\\ \;\;\;\;\frac{t\_0}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (cbrt (* g (* a (* 0.5 a)))))
        (t_1 (/ g (* a 2.0)))
        (t_2 (/ 1.0 (cbrt (* a (/ 2.0 g))))))
   (if (<= t_1 -5e-306)
     t_2
     (if (<= t_1 1e-290)
       (/ t_0 a)
       (if (<= t_1 2e+302) t_2 (* t_0 (/ 1.0 a)))))))

C

double code(double g, double a) {
	double t_0 = cbrt((g * (a * (0.5 * a))));
	double t_1 = g / (a * 2.0);
	double t_2 = 1.0 / cbrt((a * (2.0 / g)));
	double tmp;
	if (t_1 <= -5e-306) {
		tmp = t_2;
	} else if (t_1 <= 1e-290) {
		tmp = t_0 / a;
	} else if (t_1 <= 2e+302) {
		tmp = t_2;
	} else {
		tmp = t_0 * (1.0 / a);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g * (a * (0.5 * a))));
	double t_1 = g / (a * 2.0);
	double t_2 = 1.0 / Math.cbrt((a * (2.0 / g)));
	double tmp;
	if (t_1 <= -5e-306) {
		tmp = t_2;
	} else if (t_1 <= 1e-290) {
		tmp = t_0 / a;
	} else if (t_1 <= 2e+302) {
		tmp = t_2;
	} else {
		tmp = t_0 * (1.0 / a);
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = cbrt(Float64(g * Float64(a * Float64(0.5 * a))))
	t_1 = Float64(g / Float64(a * 2.0))
	t_2 = Float64(1.0 / cbrt(Float64(a * Float64(2.0 / g))))
	tmp = 0.0
	if (t_1 <= -5e-306)
		tmp = t_2;
	elseif (t_1 <= 1e-290)
		tmp = Float64(t_0 / a);
	elseif (t_1 <= 2e+302)
		tmp = t_2;
	else
		tmp = Float64(t_0 * Float64(1.0 / a));
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[Power[N[(g * N[(a * N[(0.5 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Power[N[(a * N[(2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-306], t$95$2, If[LessEqual[t$95$1, 1e-290], N[(t$95$0 / a), $MachinePrecision], If[LessEqual[t$95$1, 2e+302], t$95$2, N[(t$95$0 * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -4.99999999999999998e-306 or 1.0000000000000001e-290 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 2.0000000000000002e302

   1. Initial program 93.1%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. 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 98.8%

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

   8. Applied rewrites 43.3%

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

   10. Applied rewrites 93.9%

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

   if -4.99999999999999998e-306 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 1.0000000000000001e-290

   1. Initial program 15.8%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. 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%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. cbrt-undiv N/A

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

      7. lift-neg.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-inverses N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. /-rgt-identity N/A

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

      19. associate-/r/ N/A

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

      20. *-inverses N/A

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

      21. times-frac N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. associate-*l/ N/A

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

   8. Applied rewrites 32.9%

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

   if 2.0000000000000002e302 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 4.4%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. 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. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 99.3

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 98.4%

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

   8. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\sqrt[3]{g \cdot \left(a \cdot \left(a \cdot 0.5\right)\right)} \cdot \frac{1}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 10^{-290}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot \left(a \cdot \left(0.5 \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot \left(a \cdot \left(0.5 \cdot a\right)\right)} \cdot \frac{1}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt[3]{g \cdot \left(a \cdot \left(0.5 \cdot a\right)\right)}}{a}\\ t_1 := \frac{g}{a \cdot 2}\\ t_2 := \frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-306}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-290}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ (cbrt (* g (* a (* 0.5 a)))) a))
        (t_1 (/ g (* a 2.0)))
        (t_2 (/ 1.0 (cbrt (* a (/ 2.0 g))))))
   (if (<= t_1 -5e-306)
     t_2
     (if (<= t_1 1e-290) t_0 (if (<= t_1 2e+302) t_2 t_0)))))

C

double code(double g, double a) {
	double t_0 = cbrt((g * (a * (0.5 * a)))) / a;
	double t_1 = g / (a * 2.0);
	double t_2 = 1.0 / cbrt((a * (2.0 / g)));
	double tmp;
	if (t_1 <= -5e-306) {
		tmp = t_2;
	} else if (t_1 <= 1e-290) {
		tmp = t_0;
	} else if (t_1 <= 2e+302) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g * (a * (0.5 * a)))) / a;
	double t_1 = g / (a * 2.0);
	double t_2 = 1.0 / Math.cbrt((a * (2.0 / g)));
	double tmp;
	if (t_1 <= -5e-306) {
		tmp = t_2;
	} else if (t_1 <= 1e-290) {
		tmp = t_0;
	} else if (t_1 <= 2e+302) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(cbrt(Float64(g * Float64(a * Float64(0.5 * a)))) / a)
	t_1 = Float64(g / Float64(a * 2.0))
	t_2 = Float64(1.0 / cbrt(Float64(a * Float64(2.0 / g))))
	tmp = 0.0
	if (t_1 <= -5e-306)
		tmp = t_2;
	elseif (t_1 <= 1e-290)
		tmp = t_0;
	elseif (t_1 <= 2e+302)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(N[Power[N[(g * N[(a * N[(0.5 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$1 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Power[N[(a * N[(2.0 / g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-306], t$95$2, If[LessEqual[t$95$1, 1e-290], t$95$0, If[LessEqual[t$95$1, 2e+302], t$95$2, t$95$0]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -4.99999999999999998e-306 or 1.0000000000000001e-290 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 2.0000000000000002e302

   1. Initial program 93.1%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. 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 98.8%

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

   8. Applied rewrites 43.3%

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

   10. Applied rewrites 93.9%

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

   if -4.99999999999999998e-306 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 1.0000000000000001e-290 or 2.0000000000000002e302 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 13.4%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. 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. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. 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 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. cbrt-undiv N/A

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

      7. lift-neg.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-inverses N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. /-rgt-identity N/A

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

      19. associate-/r/ N/A

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

      20. *-inverses N/A

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

      21. times-frac N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. associate-*l/ N/A

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

   8. Applied rewrites 35.7%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -5 \cdot 10^{-306}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 10^{-290}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot \left(a \cdot \left(0.5 \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a \cdot \frac{2}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot \left(a \cdot \left(0.5 \cdot a\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+302], N[Power[t$95$0, 1/3], $MachinePrecision], N[(N[Power[N[(g * N[(a * N[(0.5 * 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)) < 2.0000000000000002e302

   1. Initial program 81.2%

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

   if 2.0000000000000002e302 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 4.4%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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. 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. cbrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 99.3

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. cbrt-undiv N/A

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

      7. lift-neg.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-inverses N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. /-rgt-identity N/A

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

      19. associate-/r/ N/A

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

      20. *-inverses N/A

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

      21. times-frac N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. associate-*l/ N/A

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

   8. Applied rewrites 46.2%

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

4. Final simplification 79.8%

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

Alternative 10: 76.1% 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 78.2%

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

3. Final simplification 78.2%

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

Alternative 11: 76.1% 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 78.2%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
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 78.1

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

4. Applied rewrites 78.1%

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

5. Final simplification 78.1%

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

Reproduce

herbie shell --seed 2024226 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))