2-ancestry mixing, zero discriminant

Percentage Accurate: 75.7% → 98.6%

Time: 9.9s

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: 75.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. cbrt-div N/A

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

   2. /-lowering-/.f64 N/A

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

   3. cbrt-lowering-cbrt.f64 N/A

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

   4. pow1/3 N/A

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

   5. pow-lowering-pow.f64 N/A

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

   6. +-lft-identity N/A

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(0 + 2 \cdot a\right)}}^{\frac{1}{3}}} \]

   7. +-commutative N/A

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(2 \cdot a + 0\right)}}^{\frac{1}{3}}} \]

   8. accelerator-lowering-fma.f64 44.4

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(\mathsf{fma}\left(2, a, 0\right)\right)}}^{0.3333333333333333}} \]

4. Applied egg-rr 44.4%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{{\left(\mathsf{fma}\left(2, a, 0\right)\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{2 \cdot a + 0}}} \]

   2. +-rgt-identity N/A

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

   3. cbrt-lowering-cbrt.f64 N/A

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

   4. +-rgt-identity N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{2 \cdot a + 0}}} \]

   5. accelerator-lowering-fma.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{\mathsf{fma}\left(2, a, 0\right)}}} \]

6. Applied egg-rr 98.7%

   \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{\mathsf{fma}\left(2, a, 0\right)}}} \]

8. Applied egg-rr 44.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{2}{\frac{\sqrt[3]{\mathsf{fma}\left(g, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 4, 0\right), 0\right), 0\right)}}{a}}}} \]

10. Applied egg-rr 98.7%

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

Alternative 2: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{\sqrt[3]{\frac{a \cdot g}{\frac{0.25}{a}}}}{\mathsf{fma}\left(2, a, 0\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+280}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\sqrt[3]{\frac{g}{\frac{2}{\frac{1}{a}}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-232}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{2}{\frac{g}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0)))
        (t_1 (/ (cbrt (/ (* a g) (/ 0.25 a))) (fma 2.0 a 0.0))))
   (if (<= t_0 -2e+280)
     t_1
     (if (<= t_0 -4e-303)
       (cbrt (/ g (/ 2.0 (/ 1.0 a))))
       (if (<= t_0 5e-232)
         t_1
         (if (<= t_0 5e+280) (cbrt (/ 1.0 (/ 2.0 (/ g a)))) t_1))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = cbrt(((a * g) / (0.25 / a))) / fma(2.0, a, 0.0);
	double tmp;
	if (t_0 <= -2e+280) {
		tmp = t_1;
	} else if (t_0 <= -4e-303) {
		tmp = cbrt((g / (2.0 / (1.0 / a))));
	} else if (t_0 <= 5e-232) {
		tmp = t_1;
	} else if (t_0 <= 5e+280) {
		tmp = cbrt((1.0 / (2.0 / (g / a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(cbrt(Float64(Float64(a * g) / Float64(0.25 / a))) / fma(2.0, a, 0.0))
	tmp = 0.0
	if (t_0 <= -2e+280)
		tmp = t_1;
	elseif (t_0 <= -4e-303)
		tmp = cbrt(Float64(g / Float64(2.0 / Float64(1.0 / a))));
	elseif (t_0 <= 5e-232)
		tmp = t_1;
	elseif (t_0 <= 5e+280)
		tmp = cbrt(Float64(1.0 / Float64(2.0 / Float64(g / a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[(a * g), $MachinePrecision] / N[(0.25 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(2.0 * a + 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+280], t$95$1, If[LessEqual[t$95$0, -4e-303], N[Power[N[(g / N[(2.0 / N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t$95$0, 5e-232], t$95$1, If[LessEqual[t$95$0, 5e+280], N[Power[N[(1.0 / N[(2.0 / N[(g / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -2.0000000000000001e280 or -3.99999999999999972e-303 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 4.9999999999999999e-232 or 5.0000000000000002e280 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 17.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. cbrt-prod N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. cbrt-prod N/A

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

      7. associate-/r* N/A

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

      8. pow1/3 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. pow-lowering-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. cbrt-lowering-cbrt.f64 N/A

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

      18. cbrt-lowering-cbrt.f64 98.6

          \[\leadsto \frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

   4. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]

   6. Applied egg-rr 38.6%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \mathsf{fma}\left(a \cdot a, 4, 0\right)}}{\mathsf{fma}\left(2, a, 0\right)}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot 4\right)}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\left(a \cdot \left(a \cdot 4\right)\right)}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(g \cdot a\right) \cdot \left(a \cdot 4\right)}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      4. remove-double-div N/A

         \[\leadsto \frac{\sqrt[3]{\left(g \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{1}{a \cdot 4}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      5. --rgt-identity N/A

         \[\leadsto \frac{\sqrt[3]{\left(g \cdot a\right) \cdot \frac{1}{\frac{1}{\color{blue}{a \cdot 4 - 0}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      6. un-div-inv N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g \cdot a}{\frac{1}{a \cdot 4 - 0}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g \cdot a}{\frac{1}{a \cdot 4 - 0}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{\frac{\color{blue}{g \cdot a}}{\frac{1}{a \cdot 4 - 0}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      9. --rgt-identity N/A

         \[\leadsto \frac{\sqrt[3]{\frac{g \cdot a}{\frac{1}{\color{blue}{a \cdot 4}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sqrt[3]{\frac{g \cdot a}{\frac{1}{a \cdot \color{blue}{\frac{1}{\frac{1}{4}}}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      11. div-inv N/A

          \[\leadsto \frac{\sqrt[3]{\frac{g \cdot a}{\frac{1}{\color{blue}{\frac{a}{\frac{1}{4}}}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      12. clear-num N/A

          \[\leadsto \frac{\sqrt[3]{\frac{g \cdot a}{\color{blue}{\frac{\frac{1}{4}}{a}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      13. /-lowering-/.f64 73.4

          \[\leadsto \frac{\sqrt[3]{\frac{g \cdot a}{\color{blue}{\frac{0.25}{a}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

   8. Applied egg-rr 73.4%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g \cdot a}{\frac{0.25}{a}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

   if -2.0000000000000001e280 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -3.99999999999999972e-303

   1. Initial program 99.1%

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

      1. metadata-eval N/A

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

      2. associate-/r/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 99.2

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

   4. Applied egg-rr 99.2%

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

   if 4.9999999999999999e-232 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 5.0000000000000002e280

   1. Initial program 99.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

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

      6. metadata-eval 99.3

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

   4. Applied egg-rr 99.3%

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

   6. Applied egg-rr 41.6%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{\mathsf{fma}\left(a, \left(a \cdot a\right) \cdot -8, 0\right)} \cdot g}{\frac{0.25}{a \cdot a}}}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{a \cdot \left(\left(a \cdot a\right) \cdot -8\right) + 0} \cdot \frac{g}{\frac{\frac{1}{4}}{a \cdot a}}}} \]

      2. frac-2neg N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(a \cdot \left(\left(a \cdot a\right) \cdot -8\right) + 0\right)\right)}} \cdot \frac{g}{\frac{\frac{1}{4}}{a \cdot a}}} \]

      3. frac-times N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\left(\mathsf{neg}\left(-1\right)\right) \cdot g}{\left(\mathsf{neg}\left(\left(a \cdot \left(\left(a \cdot a\right) \cdot -8\right) + 0\right)\right)\right) \cdot \frac{\frac{1}{4}}{a \cdot a}}}} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{1} \cdot g}{\left(\mathsf{neg}\left(\left(a \cdot \left(\left(a \cdot a\right) \cdot -8\right) + 0\right)\right)\right) \cdot \frac{\frac{1}{4}}{a \cdot a}}} \]

      5. *-lft-identity N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{g}}{\left(\mathsf{neg}\left(\left(a \cdot \left(\left(a \cdot a\right) \cdot -8\right) + 0\right)\right)\right) \cdot \frac{\frac{1}{4}}{a \cdot a}}} \]

      6. +-rgt-identity N/A

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

      7. associate-*r* N/A

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

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

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

      9. cube-unmult N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. *-commutative N/A

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

      14. +-rgt-identity N/A

          \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\left({\left(2 \cdot a\right)}^{3} + 0\right)} \cdot \frac{\frac{1}{4}}{a \cdot a}}} \]

      15. metadata-eval N/A

          \[\leadsto \sqrt[3]{\frac{g}{\left({\left(2 \cdot a\right)}^{3} + \color{blue}{{0}^{3}}\right) \cdot \frac{\frac{1}{4}}{a \cdot a}}} \]

      16. clear-num N/A

          \[\leadsto \sqrt[3]{\frac{g}{\left({\left(2 \cdot a\right)}^{3} + {0}^{3}\right) \cdot \color{blue}{\frac{1}{\frac{a \cdot a}{\frac{1}{4}}}}}} \]

      17. div-inv N/A

          \[\leadsto \sqrt[3]{\frac{g}{\left({\left(2 \cdot a\right)}^{3} + {0}^{3}\right) \cdot \frac{1}{\color{blue}{\left(a \cdot a\right) \cdot \frac{1}{\frac{1}{4}}}}}} \]

      18. metadata-eval N/A

          \[\leadsto \sqrt[3]{\frac{g}{\left({\left(2 \cdot a\right)}^{3} + {0}^{3}\right) \cdot \frac{1}{\left(a \cdot a\right) \cdot \color{blue}{4}}}} \]

      19. div-inv N/A

          \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\frac{{\left(2 \cdot a\right)}^{3} + {0}^{3}}{\left(a \cdot a\right) \cdot 4}}}} \]

   8. Applied egg-rr 99.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -2 \cdot 10^{+280}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{a \cdot g}{\frac{0.25}{a}}}}{\mathsf{fma}\left(2, a, 0\right)}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\sqrt[3]{\frac{g}{\frac{2}{\frac{1}{a}}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 5 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{a \cdot g}{\frac{0.25}{a}}}}{\mathsf{fma}\left(2, a, 0\right)}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{2}{\frac{g}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{a \cdot g}{\frac{0.25}{a}}}}{\mathsf{fma}\left(2, a, 0\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{g}{a \cdot 2}\\ t_1 := \frac{\sqrt[3]{a \cdot \left(g \cdot \mathsf{fma}\left(a, 4, 0\right)\right)}}{\mathsf{fma}\left(2, a, 0\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+280}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\sqrt[3]{\frac{g}{\frac{2}{\frac{1}{a}}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-232}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{2}{\frac{g}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (let* ((t_0 (/ g (* a 2.0)))
        (t_1 (/ (cbrt (* a (* g (fma a 4.0 0.0)))) (fma 2.0 a 0.0))))
   (if (<= t_0 -2e+280)
     t_1
     (if (<= t_0 -4e-303)
       (cbrt (/ g (/ 2.0 (/ 1.0 a))))
       (if (<= t_0 5e-232)
         t_1
         (if (<= t_0 5e+280) (cbrt (/ 1.0 (/ 2.0 (/ g a)))) t_1))))))

C

double code(double g, double a) {
	double t_0 = g / (a * 2.0);
	double t_1 = cbrt((a * (g * fma(a, 4.0, 0.0)))) / fma(2.0, a, 0.0);
	double tmp;
	if (t_0 <= -2e+280) {
		tmp = t_1;
	} else if (t_0 <= -4e-303) {
		tmp = cbrt((g / (2.0 / (1.0 / a))));
	} else if (t_0 <= 5e-232) {
		tmp = t_1;
	} else if (t_0 <= 5e+280) {
		tmp = cbrt((1.0 / (2.0 / (g / a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(g, a)
	t_0 = Float64(g / Float64(a * 2.0))
	t_1 = Float64(cbrt(Float64(a * Float64(g * fma(a, 4.0, 0.0)))) / fma(2.0, a, 0.0))
	tmp = 0.0
	if (t_0 <= -2e+280)
		tmp = t_1;
	elseif (t_0 <= -4e-303)
		tmp = cbrt(Float64(g / Float64(2.0 / Float64(1.0 / a))));
	elseif (t_0 <= 5e-232)
		tmp = t_1;
	elseif (t_0 <= 5e+280)
		tmp = cbrt(Float64(1.0 / Float64(2.0 / Float64(g / a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[g_, a_] := Block[{t$95$0 = N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(a * N[(g * N[(a * 4.0 + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(2.0 * a + 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+280], t$95$1, If[LessEqual[t$95$0, -4e-303], N[Power[N[(g / N[(2.0 / N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t$95$0, 5e-232], t$95$1, If[LessEqual[t$95$0, 5e+280], N[Power[N[(1.0 / N[(2.0 / N[(g / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -2.0000000000000001e280 or -3.99999999999999972e-303 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 4.9999999999999999e-232 or 5.0000000000000002e280 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 17.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. cbrt-prod N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. cbrt-prod N/A

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

      7. associate-/r* N/A

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

      8. pow1/3 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. pow-lowering-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. cbrt-lowering-cbrt.f64 N/A

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

      18. cbrt-lowering-cbrt.f64 98.6

          \[\leadsto \frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

   4. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]

   6. Applied egg-rr 38.6%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \mathsf{fma}\left(a \cdot a, 4, 0\right)}}{\mathsf{fma}\left(2, a, 0\right)}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot 4\right)}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{1}{4}}}\right)}}{\mathsf{fma}\left(2, a, 0\right)} \]

      3. div-inv N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\frac{a \cdot a}{\frac{1}{4}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      4. clear-num N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\frac{1}{\frac{\frac{1}{4}}{a \cdot a}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      5. div-inv N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g}{\frac{\frac{1}{4}}{a \cdot a}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\sqrt[3]{\frac{g}{\color{blue}{\frac{\frac{\frac{1}{4}}{a}}{a}}}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      7. associate-/r/ N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g}{\frac{\frac{1}{4}}{a}} \cdot a}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g}{\frac{\frac{1}{4}}{a}} \cdot a}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      9. div-inv N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(g \cdot \frac{1}{\frac{\frac{1}{4}}{a}}\right)} \cdot a}}{\mathsf{fma}\left(2, a, 0\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{\sqrt[3]{\left(g \cdot \color{blue}{\frac{a}{\frac{1}{4}}}\right) \cdot a}}{\mathsf{fma}\left(2, a, 0\right)} \]

      11. div-inv N/A

          \[\leadsto \frac{\sqrt[3]{\left(g \cdot \color{blue}{\left(a \cdot \frac{1}{\frac{1}{4}}\right)}\right) \cdot a}}{\mathsf{fma}\left(2, a, 0\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sqrt[3]{\left(g \cdot \left(a \cdot \color{blue}{4}\right)\right) \cdot a}}{\mathsf{fma}\left(2, a, 0\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(g \cdot \left(a \cdot 4\right)\right)} \cdot a}}{\mathsf{fma}\left(2, a, 0\right)} \]

      14. +-rgt-identity N/A

          \[\leadsto \frac{\sqrt[3]{\left(g \cdot \color{blue}{\left(a \cdot 4 + 0\right)}\right) \cdot a}}{\mathsf{fma}\left(2, a, 0\right)} \]

      15. accelerator-lowering-fma.f64 73.3

          \[\leadsto \frac{\sqrt[3]{\left(g \cdot \color{blue}{\mathsf{fma}\left(a, 4, 0\right)}\right) \cdot a}}{\mathsf{fma}\left(2, a, 0\right)} \]

   8. Applied egg-rr 73.3%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(g \cdot \mathsf{fma}\left(a, 4, 0\right)\right) \cdot a}}}{\mathsf{fma}\left(2, a, 0\right)} \]

   if -2.0000000000000001e280 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -3.99999999999999972e-303

   1. Initial program 99.1%

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

      1. metadata-eval N/A

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

      2. associate-/r/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 99.2

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

   4. Applied egg-rr 99.2%

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

   if 4.9999999999999999e-232 < (/.f64 g (*.f64 #s(literal 2 binary64) a)) < 5.0000000000000002e280

   1. Initial program 99.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

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

      6. metadata-eval 99.3

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

   4. Applied egg-rr 99.3%

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

   6. Applied egg-rr 41.6%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{-1}{\mathsf{fma}\left(a, \left(a \cdot a\right) \cdot -8, 0\right)} \cdot g}{\frac{0.25}{a \cdot a}}}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{-1}{a \cdot \left(\left(a \cdot a\right) \cdot -8\right) + 0} \cdot \frac{g}{\frac{\frac{1}{4}}{a \cdot a}}}} \]

      2. frac-2neg N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(a \cdot \left(\left(a \cdot a\right) \cdot -8\right) + 0\right)\right)}} \cdot \frac{g}{\frac{\frac{1}{4}}{a \cdot a}}} \]

      3. frac-times N/A

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\left(\mathsf{neg}\left(-1\right)\right) \cdot g}{\left(\mathsf{neg}\left(\left(a \cdot \left(\left(a \cdot a\right) \cdot -8\right) + 0\right)\right)\right) \cdot \frac{\frac{1}{4}}{a \cdot a}}}} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{1} \cdot g}{\left(\mathsf{neg}\left(\left(a \cdot \left(\left(a \cdot a\right) \cdot -8\right) + 0\right)\right)\right) \cdot \frac{\frac{1}{4}}{a \cdot a}}} \]

      5. *-lft-identity N/A

         \[\leadsto \sqrt[3]{\frac{\color{blue}{g}}{\left(\mathsf{neg}\left(\left(a \cdot \left(\left(a \cdot a\right) \cdot -8\right) + 0\right)\right)\right) \cdot \frac{\frac{1}{4}}{a \cdot a}}} \]

      6. +-rgt-identity N/A

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

      7. associate-*r* N/A

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

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

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

      9. cube-unmult N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. *-commutative N/A

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

      14. +-rgt-identity N/A

          \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\left({\left(2 \cdot a\right)}^{3} + 0\right)} \cdot \frac{\frac{1}{4}}{a \cdot a}}} \]

      15. metadata-eval N/A

          \[\leadsto \sqrt[3]{\frac{g}{\left({\left(2 \cdot a\right)}^{3} + \color{blue}{{0}^{3}}\right) \cdot \frac{\frac{1}{4}}{a \cdot a}}} \]

      16. clear-num N/A

          \[\leadsto \sqrt[3]{\frac{g}{\left({\left(2 \cdot a\right)}^{3} + {0}^{3}\right) \cdot \color{blue}{\frac{1}{\frac{a \cdot a}{\frac{1}{4}}}}}} \]

      17. div-inv N/A

          \[\leadsto \sqrt[3]{\frac{g}{\left({\left(2 \cdot a\right)}^{3} + {0}^{3}\right) \cdot \frac{1}{\color{blue}{\left(a \cdot a\right) \cdot \frac{1}{\frac{1}{4}}}}}} \]

      18. metadata-eval N/A

          \[\leadsto \sqrt[3]{\frac{g}{\left({\left(2 \cdot a\right)}^{3} + {0}^{3}\right) \cdot \frac{1}{\left(a \cdot a\right) \cdot \color{blue}{4}}}} \]

      19. div-inv N/A

          \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{\frac{{\left(2 \cdot a\right)}^{3} + {0}^{3}}{\left(a \cdot a\right) \cdot 4}}}} \]

   8. Applied egg-rr 99.5%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -2 \cdot 10^{+280}:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(g \cdot \mathsf{fma}\left(a, 4, 0\right)\right)}}{\mathsf{fma}\left(2, a, 0\right)}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\sqrt[3]{\frac{g}{\frac{2}{\frac{1}{a}}}}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 5 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(g \cdot \mathsf{fma}\left(a, 4, 0\right)\right)}}{\mathsf{fma}\left(2, a, 0\right)}\\ \mathbf{elif}\;\frac{g}{a \cdot 2} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{2}{\frac{g}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a \cdot \left(g \cdot \mathsf{fma}\left(a, 4, 0\right)\right)}}{\mathsf{fma}\left(2, a, 0\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. cbrt-div N/A

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

   2. /-lowering-/.f64 N/A

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

   3. cbrt-lowering-cbrt.f64 N/A

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

   4. pow1/3 N/A

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

   5. pow-lowering-pow.f64 N/A

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

   6. +-lft-identity N/A

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(0 + 2 \cdot a\right)}}^{\frac{1}{3}}} \]

   7. +-commutative N/A

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(2 \cdot a + 0\right)}}^{\frac{1}{3}}} \]

   8. accelerator-lowering-fma.f64 44.4

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(\mathsf{fma}\left(2, a, 0\right)\right)}}^{0.3333333333333333}} \]

4. Applied egg-rr 44.4%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{{\left(\mathsf{fma}\left(2, a, 0\right)\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{2 \cdot a + 0}}} \]

   2. +-rgt-identity N/A

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

   3. cbrt-lowering-cbrt.f64 N/A

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

   4. +-rgt-identity N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{2 \cdot a + 0}}} \]

   5. accelerator-lowering-fma.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{\mathsf{fma}\left(2, a, 0\right)}}} \]

6. Applied egg-rr 98.7%

   \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{\mathsf{fma}\left(2, a, 0\right)}}} \]
7. Step-by-step derivation

   1. cbrt-undiv N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2 \cdot a + 0}}} \]

   2. +-rgt-identity N/A

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

   3. associate-/r* N/A

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

   4. cbrt-div N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. cbrt-div N/A

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

   8. div-inv N/A

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

   9. cbrt-prod N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cbrt-lowering-cbrt.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. cbrt-lowering-cbrt.f64 N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f64 N/A

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

   16. metadata-eval 98.7

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

8. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\sqrt[3]{g \cdot 0.5} \cdot \sqrt[3]{\frac{1}{a}}} \]
9. 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 78.9%

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

   1. cbrt-div N/A

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

   2. /-lowering-/.f64 N/A

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

   3. cbrt-lowering-cbrt.f64 N/A

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

   4. pow1/3 N/A

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

   5. pow-lowering-pow.f64 N/A

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

   6. +-lft-identity N/A

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(0 + 2 \cdot a\right)}}^{\frac{1}{3}}} \]

   7. +-commutative N/A

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(2 \cdot a + 0\right)}}^{\frac{1}{3}}} \]

   8. accelerator-lowering-fma.f64 44.4

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(\mathsf{fma}\left(2, a, 0\right)\right)}}^{0.3333333333333333}} \]

4. Applied egg-rr 44.4%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{{\left(\mathsf{fma}\left(2, a, 0\right)\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{2 \cdot a + 0}}} \]

   2. +-rgt-identity N/A

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

   3. cbrt-lowering-cbrt.f64 N/A

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

   4. +-rgt-identity N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{2 \cdot a + 0}}} \]

   5. accelerator-lowering-fma.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{\mathsf{fma}\left(2, a, 0\right)}}} \]

6. Applied egg-rr 98.7%

   \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{\mathsf{fma}\left(2, a, 0\right)}}} \]
7. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 98.7

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

8. Applied egg-rr 98.7%

   \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{a \cdot 2}}} \]
9. 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 78.9%

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

   1. div-inv N/A

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

   2. cbrt-prod N/A

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

   3. pow1/3 N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. pow-lowering-pow.f64 N/A

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

   9. +-lft-identity N/A

      \[\leadsto {\color{blue}{\left(0 + 2 \cdot a\right)}}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{g} \]

   10. +-commutative N/A

       \[\leadsto {\color{blue}{\left(2 \cdot a + 0\right)}}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{g} \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(2, a, 0\right)\right)}}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \sqrt[3]{g} \]

   12. metadata-eval N/A

       \[\leadsto {\left(\mathsf{fma}\left(2, a, 0\right)\right)}^{\color{blue}{\frac{-1}{3}}} \cdot \sqrt[3]{g} \]

   13. cbrt-lowering-cbrt.f64 44.4

       \[\leadsto {\left(\mathsf{fma}\left(2, a, 0\right)\right)}^{-0.3333333333333333} \cdot \color{blue}{\sqrt[3]{g}} \]

4. Applied egg-rr 44.4%

   \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(2, a, 0\right)\right)}^{-0.3333333333333333} \cdot \sqrt[3]{g}} \]
5. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. clear-num N/A

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

   8. pow1/3 N/A

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

   9. cbrt-lowering-cbrt.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. associate-/r* N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f64 98.7

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 7: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -2 \cdot 10^{+280}:\\ \;\;\;\;\frac{\sqrt[3]{4 \cdot \left(g \cdot \left(a \cdot a\right)\right)}}{\mathsf{fma}\left(2, a, 0\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g \cdot 0.125}{a \cdot 0.25}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g a)
 :precision binary64
 (if (<= (/ g (* a 2.0)) -2e+280)
   (/ (cbrt (* 4.0 (* g (* a a)))) (fma 2.0 a 0.0))
   (cbrt (/ (* g 0.125) (* a 0.25)))))

C

double code(double g, double a) {
	double tmp;
	if ((g / (a * 2.0)) <= -2e+280) {
		tmp = cbrt((4.0 * (g * (a * a)))) / fma(2.0, a, 0.0);
	} else {
		tmp = cbrt(((g * 0.125) / (a * 0.25)));
	}
	return tmp;
}

Julia

function code(g, a)
	tmp = 0.0
	if (Float64(g / Float64(a * 2.0)) <= -2e+280)
		tmp = Float64(cbrt(Float64(4.0 * Float64(g * Float64(a * a)))) / fma(2.0, a, 0.0));
	else
		tmp = cbrt(Float64(Float64(g * 0.125) / Float64(a * 0.25)));
	end
	return tmp
end

Wolfram

code[g_, a_] := If[LessEqual[N[(g / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -2e+280], N[(N[Power[N[(4.0 * N[(g * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(2.0 * a + 0.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(g * 0.125), $MachinePrecision] / N[(a * 0.25), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 g (*.f64 #s(literal 2 binary64) a)) < -2.0000000000000001e280

   1. Initial program 9.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. cbrt-prod N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. cbrt-prod N/A

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

      7. associate-/r* N/A

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

      8. pow1/3 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. pow-lowering-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. cbrt-lowering-cbrt.f64 N/A

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

      18. cbrt-lowering-cbrt.f64 98.2

          \[\leadsto \frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\color{blue}{\sqrt[3]{a}}} \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\frac{{2}^{-0.3333333333333333} \cdot \sqrt[3]{g}}{\sqrt[3]{a}}} \]

   6. Applied egg-rr 45.1%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g \cdot \mathsf{fma}\left(a \cdot a, 4, 0\right)}}{\mathsf{fma}\left(2, a, 0\right)}} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot 4\right)}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(g \cdot \left(a \cdot a\right)\right) \cdot 4}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      3. /-rgt-identity N/A

         \[\leadsto \frac{\sqrt[3]{\left(g \cdot \color{blue}{\frac{a \cdot a}{1}}\right) \cdot 4}}{\mathsf{fma}\left(2, a, 0\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt[3]{\left(g \cdot \frac{a \cdot a}{\color{blue}{\mathsf{neg}\left(-1\right)}}\right) \cdot 4}}{\mathsf{fma}\left(2, a, 0\right)} \]

      5. clear-num N/A

         \[\leadsto \frac{\sqrt[3]{\left(g \cdot \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(-1\right)}{a \cdot a}}}\right) \cdot 4}}{\mathsf{fma}\left(2, a, 0\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sqrt[3]{\left(g \cdot \frac{1}{\frac{\color{blue}{1}}{a \cdot a}}\right) \cdot 4}}{\mathsf{fma}\left(2, a, 0\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g}{\frac{1}{a \cdot a}}} \cdot 4}}{\mathsf{fma}\left(2, a, 0\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g}{\frac{1}{a \cdot a}} \cdot 4}}}{\mathsf{fma}\left(2, a, 0\right)} \]

      9. div-inv N/A

         \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(g \cdot \frac{1}{\frac{1}{a \cdot a}}\right)} \cdot 4}}{\mathsf{fma}\left(2, a, 0\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sqrt[3]{\left(g \cdot \frac{1}{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{a \cdot a}}\right) \cdot 4}}{\mathsf{fma}\left(2, a, 0\right)} \]

      11. clear-num N/A

          \[\leadsto \frac{\sqrt[3]{\left(g \cdot \color{blue}{\frac{a \cdot a}{\mathsf{neg}\left(-1\right)}}\right) \cdot 4}}{\mathsf{fma}\left(2, a, 0\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sqrt[3]{\left(g \cdot \frac{a \cdot a}{\color{blue}{1}}\right) \cdot 4}}{\mathsf{fma}\left(2, a, 0\right)} \]

      13. /-rgt-identity N/A

          \[\leadsto \frac{\sqrt[3]{\left(g \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot 4}}{\mathsf{fma}\left(2, a, 0\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(g \cdot \left(a \cdot a\right)\right)} \cdot 4}}{\mathsf{fma}\left(2, a, 0\right)} \]

      15. *-lowering-*.f64 45.1

          \[\leadsto \frac{\sqrt[3]{\left(g \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot 4}}{\mathsf{fma}\left(2, a, 0\right)} \]

   8. Applied egg-rr 45.1%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(g \cdot \left(a \cdot a\right)\right) \cdot 4}}}{\mathsf{fma}\left(2, a, 0\right)} \]

   if -2.0000000000000001e280 < (/.f64 g (*.f64 #s(literal 2 binary64) a))

   1. Initial program 84.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

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

      6. metadata-eval 84.1

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

   4. Applied egg-rr 84.1%

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

      1. associate-*l/ N/A

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

      2. metadata-eval N/A

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

      3. associate-/r/ N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. unpow1 N/A

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

      7. metadata-eval N/A

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

      8. pow-div N/A

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

      9. pow3 N/A

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

      10. pow2 N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

      13. *-commutative N/A

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

      14. +-rgt-identity N/A

          \[\leadsto \sqrt[3]{\frac{1}{\frac{\frac{8 \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\color{blue}{\left(a \cdot a\right) \cdot 4 + 0}}}{g}}} \]

      15. clear-num N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\frac{8 \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\left(a \cdot a\right) \cdot 4 + 0}}}} \]

      16. +-rgt-identity N/A

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

      17. associate-/l* N/A

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

      18. associate-/r* N/A

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

   6. Applied egg-rr 84.2%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g \cdot 0.125}{a \cdot 0.25}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{g}{a \cdot 2} \leq -2 \cdot 10^{+280}:\\ \;\;\;\;\frac{\sqrt[3]{4 \cdot \left(g \cdot \left(a \cdot a\right)\right)}}{\mathsf{fma}\left(2, a, 0\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g \cdot 0.125}{a \cdot 0.25}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. cbrt-div N/A

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

   2. /-lowering-/.f64 N/A

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

   3. cbrt-lowering-cbrt.f64 N/A

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

   4. pow1/3 N/A

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

   5. pow-lowering-pow.f64 N/A

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

   6. +-lft-identity N/A

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(0 + 2 \cdot a\right)}}^{\frac{1}{3}}} \]

   7. +-commutative N/A

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(2 \cdot a + 0\right)}}^{\frac{1}{3}}} \]

   8. accelerator-lowering-fma.f64 44.4

      \[\leadsto \frac{\sqrt[3]{g}}{{\color{blue}{\left(\mathsf{fma}\left(2, a, 0\right)\right)}}^{0.3333333333333333}} \]

4. Applied egg-rr 44.4%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{{\left(\mathsf{fma}\left(2, a, 0\right)\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. unpow1/3 N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{2 \cdot a + 0}}} \]

   2. +-rgt-identity N/A

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

   3. cbrt-lowering-cbrt.f64 N/A

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

   4. +-rgt-identity N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{2 \cdot a + 0}}} \]

   5. accelerator-lowering-fma.f64 98.7

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{\mathsf{fma}\left(2, a, 0\right)}}} \]

6. Applied egg-rr 98.7%

   \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{\mathsf{fma}\left(2, a, 0\right)}}} \]

8. Applied egg-rr 44.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{2}{\frac{\sqrt[3]{\mathsf{fma}\left(g, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 4, 0\right), 0\right), 0\right)}}{a}}}} \]

10. Applied egg-rr 79.6%

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

Alternative 9: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g \cdot 0.125}{a \cdot 0.25}} \end{array} \]

FPCore

(FPCore (g a) :precision binary64 (cbrt (/ (* g 0.125) (* a 0.25))))

C

double code(double g, double a) {
	return cbrt(((g * 0.125) / (a * 0.25)));
}

Java

public static double code(double g, double a) {
	return Math.cbrt(((g * 0.125) / (a * 0.25)));
}

Julia

function code(g, a)
	return cbrt(Float64(Float64(g * 0.125) / Float64(a * 0.25)))
end

Wolfram

code[g_, a_] := N[Power[N[(N[(g * 0.125), $MachinePrecision] / N[(a * 0.25), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 78.9%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. *-lowering-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

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

   6. metadata-eval 78.9

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

4. Applied egg-rr 78.9%

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

   1. associate-*l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-/r/ N/A

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

   4. associate-/r* N/A

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

   5. associate-*l/ N/A

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

   6. unpow1 N/A

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

   7. metadata-eval N/A

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

   8. pow-div N/A

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

   9. pow3 N/A

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

   10. pow2 N/A

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

   11. metadata-eval N/A

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

   12. times-frac N/A

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

   13. *-commutative N/A

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

   14. +-rgt-identity N/A

       \[\leadsto \sqrt[3]{\frac{1}{\frac{\frac{8 \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\color{blue}{\left(a \cdot a\right) \cdot 4 + 0}}}{g}}} \]

   15. clear-num N/A

       \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{\frac{8 \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\left(a \cdot a\right) \cdot 4 + 0}}}} \]

   16. +-rgt-identity N/A

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

   17. associate-/l* N/A

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

   18. associate-/r* N/A

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

6. Applied egg-rr 78.9%

   \[\leadsto \sqrt[3]{\color{blue}{\frac{g \cdot 0.125}{a \cdot 0.25}}} \]
7. Add Preprocessing

Alternative 10: 75.7% 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 78.9%

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

3. Final simplification 78.9%

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

Alternative 11: 75.7% 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 78.9%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. *-lowering-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

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

   6. metadata-eval 78.9

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

4. Applied egg-rr 78.9%

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

5. Final simplification 78.9%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))