2-ancestry mixing, zero discriminant

Percentage Accurate: 76.1% → 98.7%

Time: 6.9s

Alternatives: 11

Speedup: 0.8×

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 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. 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%

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

Alternative 2: 90.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) -2e-291)
   (* (pow (* a -2.0) -0.3333333333333333) (cbrt (- g)))
   (* (cbrt (* g 0.5)) (pow a -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -2e-291) {
		tmp = pow((a * -2.0), -0.3333333333333333) * cbrt(-g);
	} else {
		tmp = cbrt((g * 0.5)) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= -2e-291) {
		tmp = Math.pow((a * -2.0), -0.3333333333333333) * Math.cbrt(-g);
	} else {
		tmp = Math.cbrt((g * 0.5)) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= -2e-291)
		tmp = Float64((Float64(a * -2.0) ^ -0.3333333333333333) * cbrt(Float64(-g)));
	else
		tmp = Float64(cbrt(Float64(g * 0.5)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], -2e-291], N[(N[Power[N[(a * -2.0), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[(-g), 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) a) < -1.99999999999999992e-291

   1. Initial program 72.0%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 36.5%

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

   8. Applied rewrites 91.9%

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

   if -1.99999999999999992e-291 < (*.f64 #s(literal 2 binary64) a)

   1. Initial program 75.6%

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

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-cbrt.f64 N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 92.2

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

   4. Applied rewrites 92.2%

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

4. Final simplification 92.0%

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

Alternative 3: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 5 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (* a 2.0) 5e-308)
   (/ 1.0 (cbrt (/ a (* g 0.5))))
   (* (cbrt (* g 0.5)) (pow a -0.3333333333333333))))

C

double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 5e-308) {
		tmp = 1.0 / cbrt((a / (g * 0.5)));
	} else {
		tmp = cbrt((g * 0.5)) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((a * 2.0) <= 5e-308) {
		tmp = 1.0 / Math.cbrt((a / (g * 0.5)));
	} else {
		tmp = Math.cbrt((g * 0.5)) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(a * 2.0) <= 5e-308)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))));
	else
		tmp = Float64(cbrt(Float64(g * 0.5)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(a * 2.0), $MachinePrecision], 5e-308], N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(g * 0.5), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.2%

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

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. clear-num N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 72.5

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

   4. Applied rewrites 72.5%

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

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

   1. Initial program 75.4%

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

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-cbrt.f64 N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 92.2

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

   4. Applied rewrites 92.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 5 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot 0.5} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 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. 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 5: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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. 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 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-undiv N/A

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

   5. div-inv N/A

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

   6. cbrt-prod N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. div-inv N/A

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

   12. clear-num N/A

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

   13. lower-cbrt.f64 N/A

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

   14. lower-/.f64 98.7

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

8. Applied rewrites 98.7%

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

Alternative 6: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{a}{\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}}}\\ t_1 := \frac{g}{a \cdot 2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-320}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\sqrt[3]{\frac{g}{\frac{2}{\frac{1}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ a (cbrt (* g (* 0.5 (* a a))))))) (t_1 (/ g (* a 2.0))))
   (if (<= t_1 -1e-284)
     (/ 1.0 (cbrt (/ a (* g 0.5))))
     (if (<= t_1 5e-320)
       t_0
       (if (<= t_1 4e+295) (cbrt (/ g (/ 2.0 (/ 1.0 a)))) t_0)))))

C

double code(double g, double a) {
	double t_0 = 1.0 / (a / cbrt((g * (0.5 * (a * a)))));
	double t_1 = g / (a * 2.0);
	double tmp;
	if (t_1 <= -1e-284) {
		tmp = 1.0 / cbrt((a / (g * 0.5)));
	} else if (t_1 <= 5e-320) {
		tmp = t_0;
	} else if (t_1 <= 4e+295) {
		tmp = cbrt((g / (2.0 / (1.0 / a))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = 1.0 / (a / Math.cbrt((g * (0.5 * (a * a)))));
	double t_1 = g / (a * 2.0);
	double tmp;
	if (t_1 <= -1e-284) {
		tmp = 1.0 / Math.cbrt((a / (g * 0.5)));
	} else if (t_1 <= 5e-320) {
		tmp = t_0;
	} else if (t_1 <= 4e+295) {
		tmp = Math.cbrt((g / (2.0 / (1.0 / a))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(1.0 / Float64(a / cbrt(Float64(g * Float64(0.5 * Float64(a * a))))))
	t_1 = Float64(g / Float64(a * 2.0))
	tmp = 0.0
	if (t_1 <= -1e-284)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))));
	elseif (t_1 <= 5e-320)
		tmp = t_0;
	elseif (t_1 <= 4e+295)
		tmp = cbrt(Float64(g / Float64(2.0 / Float64(1.0 / a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(1.0 / N[(a / N[Power[N[(g * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-284], N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-320], t$95$0, If[LessEqual[t$95$1, 4e+295], N[Power[N[(g / N[(2.0 / N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -1.00000000000000004e-284

   1. Initial program 83.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. clear-num N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 85.2

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

   4. Applied rewrites 85.2%

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

   if -1.00000000000000004e-284 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 4.99994e-320 or 3.9999999999999999e295 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 9.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 12.9%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      2. lift-cbrt.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{\color{blue}{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      4. cbrt-prod N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{\sqrt[3]{8} \cdot \sqrt[3]{a \cdot \left(a \cdot a\right)}}} \]

      5. pow1/3 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{{8}^{\frac{1}{3}}} \cdot \sqrt[3]{a \cdot \left(a \cdot a\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{a \cdot \left(a \cdot a\right)}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{a \cdot \color{blue}{\left(a \cdot a\right)}}} \]

      8. cube-unmult N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{{a}^{3}}}} \]

      9. rem-cbrt-cube N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \color{blue}{a}} \]

      10. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}}}{a}} \]

      11. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}}}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}}}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}}}}} \]

      14. lift-cbrt.f64 N/A

          \[\leadsto \frac{1}{\frac{a}{\frac{\color{blue}{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}}{{8}^{\frac{1}{3}}}}} \]

      15. pow1/3 N/A

          \[\leadsto \frac{1}{\frac{a}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{\sqrt[3]{8}}}}} \]

      16. cbrt-undiv N/A

          \[\leadsto \frac{1}{\frac{a}{\color{blue}{\sqrt[3]{\frac{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}{8}}}}} \]

      17. lower-cbrt.f64 N/A

          \[\leadsto \frac{1}{\frac{a}{\color{blue}{\sqrt[3]{\frac{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}{8}}}}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a}{\sqrt[3]{\frac{\color{blue}{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{8}}}} \]

      19. associate-/l* N/A

          \[\leadsto \frac{1}{\frac{a}{\sqrt[3]{\color{blue}{g \cdot \frac{\left(a \cdot a\right) \cdot 4}{8}}}}} \]

   8. Applied rewrites 33.3%

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

   if 4.99994e-320 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 3.9999999999999999e295

   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. metadata-eval N/A

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

      3. associate-/r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.0

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

   4. Applied rewrites 99.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 5 \cdot 10^{-320}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\sqrt[3]{\frac{g}{\frac{2}{\frac{1}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)} \cdot \frac{1}{a}\\ t_1 := \frac{g}{a \cdot 2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-320}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\sqrt[3]{\frac{g}{\frac{2}{\frac{1}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (* (cbrt (* g (* 0.5 (* a a)))) (/ 1.0 a))) (t_1 (/ g (* a 2.0))))
   (if (<= t_1 -1e-284)
     (/ 1.0 (cbrt (/ a (* g 0.5))))
     (if (<= t_1 5e-320)
       t_0
       (if (<= t_1 4e+295) (cbrt (/ g (/ 2.0 (/ 1.0 a)))) t_0)))))

C

double code(double g, double a) {
	double t_0 = cbrt((g * (0.5 * (a * a)))) * (1.0 / a);
	double t_1 = g / (a * 2.0);
	double tmp;
	if (t_1 <= -1e-284) {
		tmp = 1.0 / cbrt((a / (g * 0.5)));
	} else if (t_1 <= 5e-320) {
		tmp = t_0;
	} else if (t_1 <= 4e+295) {
		tmp = cbrt((g / (2.0 / (1.0 / a))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g * (0.5 * (a * a)))) * (1.0 / a);
	double t_1 = g / (a * 2.0);
	double tmp;
	if (t_1 <= -1e-284) {
		tmp = 1.0 / Math.cbrt((a / (g * 0.5)));
	} else if (t_1 <= 5e-320) {
		tmp = t_0;
	} else if (t_1 <= 4e+295) {
		tmp = Math.cbrt((g / (2.0 / (1.0 / a))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(cbrt(Float64(g * Float64(0.5 * Float64(a * a)))) * Float64(1.0 / a))
	t_1 = Float64(g / Float64(a * 2.0))
	tmp = 0.0
	if (t_1 <= -1e-284)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))));
	elseif (t_1 <= 5e-320)
		tmp = t_0;
	elseif (t_1 <= 4e+295)
		tmp = cbrt(Float64(g / Float64(2.0 / Float64(1.0 / a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(N[Power[N[(g * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-284], N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-320], t$95$0, If[LessEqual[t$95$1, 4e+295], N[Power[N[(g / N[(2.0 / N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -1.00000000000000004e-284

   1. Initial program 83.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. clear-num N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 85.2

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

   4. Applied rewrites 85.2%

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

   if -1.00000000000000004e-284 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 4.99994e-320 or 3.9999999999999999e295 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 9.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 12.9%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      2. lift-cbrt.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{\color{blue}{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      4. cbrt-prod N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{\sqrt[3]{8} \cdot \sqrt[3]{a \cdot \left(a \cdot a\right)}}} \]

      5. pow1/3 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{{8}^{\frac{1}{3}}} \cdot \sqrt[3]{a \cdot \left(a \cdot a\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{a \cdot \left(a \cdot a\right)}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{a \cdot \color{blue}{\left(a \cdot a\right)}}} \]

      8. cube-unmult N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{{a}^{3}}}} \]

      9. rem-cbrt-cube N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \color{blue}{a}} \]

      10. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}}}{a}} \]

      11. div-inv N/A

          \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}} \cdot \frac{1}{a}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}} \cdot \frac{1}{a}} \]

   8. Applied rewrites 33.3%

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

   if 4.99994e-320 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 3.9999999999999999e295

   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. metadata-eval N/A

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

      3. associate-/r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.0

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

   4. Applied rewrites 99.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 5 \cdot 10^{-320}:\\ \;\;\;\;\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)} \cdot \frac{1}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\sqrt[3]{\frac{g}{\frac{2}{\frac{1}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)} \cdot \frac{1}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)} \cdot \frac{1}{a}\\ t_1 := \frac{g}{a \cdot 2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-320}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (* (cbrt (* g (* 0.5 (* a a)))) (/ 1.0 a))) (t_1 (/ g (* a 2.0))))
   (if (<= t_1 -1e-284)
     (/ 1.0 (cbrt (/ a (* g 0.5))))
     (if (<= t_1 5e-320)
       t_0
       (if (<= t_1 4e+295) (cbrt (* g (/ 0.5 a))) t_0)))))

C

double code(double g, double a) {
	double t_0 = cbrt((g * (0.5 * (a * a)))) * (1.0 / a);
	double t_1 = g / (a * 2.0);
	double tmp;
	if (t_1 <= -1e-284) {
		tmp = 1.0 / cbrt((a / (g * 0.5)));
	} else if (t_1 <= 5e-320) {
		tmp = t_0;
	} else if (t_1 <= 4e+295) {
		tmp = cbrt((g * (0.5 / a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g * (0.5 * (a * a)))) * (1.0 / a);
	double t_1 = g / (a * 2.0);
	double tmp;
	if (t_1 <= -1e-284) {
		tmp = 1.0 / Math.cbrt((a / (g * 0.5)));
	} else if (t_1 <= 5e-320) {
		tmp = t_0;
	} else if (t_1 <= 4e+295) {
		tmp = Math.cbrt((g * (0.5 / a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(cbrt(Float64(g * Float64(0.5 * Float64(a * a)))) * Float64(1.0 / a))
	t_1 = Float64(g / Float64(a * 2.0))
	tmp = 0.0
	if (t_1 <= -1e-284)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))));
	elseif (t_1 <= 5e-320)
		tmp = t_0;
	elseif (t_1 <= 4e+295)
		tmp = cbrt(Float64(g * Float64(0.5 / a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(N[Power[N[(g * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-284], N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-320], t$95$0, If[LessEqual[t$95$1, 4e+295], N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -1.00000000000000004e-284

   1. Initial program 83.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. clear-num N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 85.2

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

   4. Applied rewrites 85.2%

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

   if -1.00000000000000004e-284 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 4.99994e-320 or 3.9999999999999999e295 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 9.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 12.9%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      2. lift-cbrt.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{\color{blue}{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      4. cbrt-prod N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{\sqrt[3]{8} \cdot \sqrt[3]{a \cdot \left(a \cdot a\right)}}} \]

      5. pow1/3 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{{8}^{\frac{1}{3}}} \cdot \sqrt[3]{a \cdot \left(a \cdot a\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{a \cdot \left(a \cdot a\right)}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{a \cdot \color{blue}{\left(a \cdot a\right)}}} \]

      8. cube-unmult N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{{a}^{3}}}} \]

      9. rem-cbrt-cube N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \color{blue}{a}} \]

      10. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}}}{a}} \]

      11. div-inv N/A

          \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}} \cdot \frac{1}{a}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}} \cdot \frac{1}{a}} \]

   8. Applied rewrites 33.3%

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

   if 4.99994e-320 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 3.9999999999999999e295

   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

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

      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 99.0

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

   4. Applied rewrites 99.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 5 \cdot 10^{-320}:\\ \;\;\;\;\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)} \cdot \frac{1}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)} \cdot \frac{1}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}}{a}\\ t_1 := \frac{g}{a \cdot 2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-320}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ (cbrt (* g (* 0.5 (* a a)))) a)) (t_1 (/ g (* a 2.0))))
   (if (<= t_1 -1e-284)
     (/ 1.0 (cbrt (/ a (* g 0.5))))
     (if (<= t_1 5e-320)
       t_0
       (if (<= t_1 4e+295) (cbrt (* g (/ 0.5 a))) t_0)))))

C

double code(double g, double a) {
	double t_0 = cbrt((g * (0.5 * (a * a)))) / a;
	double t_1 = g / (a * 2.0);
	double tmp;
	if (t_1 <= -1e-284) {
		tmp = 1.0 / cbrt((a / (g * 0.5)));
	} else if (t_1 <= 5e-320) {
		tmp = t_0;
	} else if (t_1 <= 4e+295) {
		tmp = cbrt((g * (0.5 / a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double t_0 = Math.cbrt((g * (0.5 * (a * a)))) / a;
	double t_1 = g / (a * 2.0);
	double tmp;
	if (t_1 <= -1e-284) {
		tmp = 1.0 / Math.cbrt((a / (g * 0.5)));
	} else if (t_1 <= 5e-320) {
		tmp = t_0;
	} else if (t_1 <= 4e+295) {
		tmp = Math.cbrt((g * (0.5 / a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(cbrt(Float64(g * Float64(0.5 * Float64(a * a)))) / a)
	t_1 = Float64(g / Float64(a * 2.0))
	tmp = 0.0
	if (t_1 <= -1e-284)
		tmp = Float64(1.0 / cbrt(Float64(a / Float64(g * 0.5))));
	elseif (t_1 <= 5e-320)
		tmp = t_0;
	elseif (t_1 <= 4e+295)
		tmp = cbrt(Float64(g * Float64(0.5 / a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(N[Power[N[(g * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$1 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-284], N[(1.0 / N[Power[N[(a / N[(g * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-320], t$95$0, If[LessEqual[t$95$1, 4e+295], N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -1.00000000000000004e-284

   1. Initial program 83.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. clear-num N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 85.2

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

   4. Applied rewrites 85.2%

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

   if -1.00000000000000004e-284 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 4.99994e-320 or 3.9999999999999999e295 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 9.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 12.9%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      2. lift-cbrt.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{\color{blue}{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      4. cbrt-prod N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{\sqrt[3]{8} \cdot \sqrt[3]{a \cdot \left(a \cdot a\right)}}} \]

      5. pow1/3 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{{8}^{\frac{1}{3}}} \cdot \sqrt[3]{a \cdot \left(a \cdot a\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{a \cdot \left(a \cdot a\right)}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{a \cdot \color{blue}{\left(a \cdot a\right)}}} \]

      8. cube-unmult N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{{a}^{3}}}} \]

      9. rem-cbrt-cube N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \color{blue}{a}} \]

      10. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}}}{a}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}}}{a}} \]

   8. Applied rewrites 33.2%

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

   if 4.99994e-320 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 3.9999999999999999e295

   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

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

      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 99.0

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

   4. Applied rewrites 99.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a}{g \cdot 0.5}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 5 \cdot 10^{-320}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}}{a}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (/ g (* a 2.0)) 4e+295)
   (cbrt (* g (/ 0.5 a)))
   (/ (cbrt (* g (* 0.5 (* a a)))) a)))

C

double code(double g, double a) {
	double tmp;
	if ((g / (a * 2.0)) <= 4e+295) {
		tmp = cbrt((g * (0.5 / a)));
	} else {
		tmp = cbrt((g * (0.5 * (a * a)))) / a;
	}
	return tmp;
}

Java

public static double code(double g, double a) {
	double tmp;
	if ((g / (a * 2.0)) <= 4e+295) {
		tmp = Math.cbrt((g * (0.5 / a)));
	} else {
		tmp = Math.cbrt((g * (0.5 * (a * a)))) / a;
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(g / Float64(a * 2.0)) <= 4e+295)
		tmp = cbrt(Float64(g * Float64(0.5 / a)));
	else
		tmp = Float64(cbrt(Float64(g * Float64(0.5 * Float64(a * a)))) / a);
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 4e+295], N[Power[N[(g * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[N[(g * N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 3.9999999999999999e295

   1. Initial program 80.1%

      \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
   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 80.1

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

   4. Applied rewrites 80.1%

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

   if 3.9999999999999999e295 < (/.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 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 7.9%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      2. lift-cbrt.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{\sqrt[3]{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\sqrt[3]{\color{blue}{8 \cdot \left(a \cdot \left(a \cdot a\right)\right)}}} \]

      4. cbrt-prod N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{\sqrt[3]{8} \cdot \sqrt[3]{a \cdot \left(a \cdot a\right)}}} \]

      5. pow1/3 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{\color{blue}{{8}^{\frac{1}{3}}} \cdot \sqrt[3]{a \cdot \left(a \cdot a\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{a \cdot \left(a \cdot a\right)}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{a \cdot \color{blue}{\left(a \cdot a\right)}}} \]

      8. cube-unmult N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{{a}^{3}}}} \]

      9. rem-cbrt-cube N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}} \cdot \color{blue}{a}} \]

      10. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}}}{a}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot 4\right)}}{{8}^{\frac{1}{3}}}}{a}} \]

   8. Applied rewrites 29.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq 4 \cdot 10^{+295}:\\ \;\;\;\;\sqrt[3]{g \cdot \frac{0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 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 73.9%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 73.9

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

4. Applied rewrites 73.9%

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

5. Final simplification 73.9%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))