2-ancestry mixing, positive discriminant

Percentage Accurate: 44.5% → 96.0%

Time: 14.7s

Alternatives: 8

Speedup: 2.6×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Initial Program: 44.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 96.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.6%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

   4. neg-lowering-neg.f64 26.2

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

5. Simplified 26.2%

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

6. Taylor expanded in g around inf

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

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

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

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

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

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

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

   4. cbrt-lowering-cbrt.f64 75.2

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

8. Simplified 75.2%

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

   1. *-commutative N/A

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

   2. cbrt-unprod N/A

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

   3. neg-mul-1 N/A

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

   4. distribute-frac-neg N/A

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

   5. cbrt-div N/A

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

   6. /-lowering-/.f64 N/A

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

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

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

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

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

   9. cbrt-lowering-cbrt.f64 96.0

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

10. Applied egg-rr 96.0%

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

    1. cbrt-undiv N/A

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

    2. neg-mul-1 N/A

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

    3. associate-/l* N/A

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

    4. div-inv N/A

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

    5. *-commutative N/A

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

    6. cbrt-unprod N/A

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

    7. pow1/3 N/A

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

    8. unpow-prod-down N/A

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

    9. associate-*r* N/A

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

    10. pow1/3 N/A

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

    11. pow-prod-down N/A

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

    12. neg-mul-1 N/A

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

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

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

    14. pow1/3 N/A

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

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

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

    16. distribute-neg-frac N/A

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

    17. metadata-eval N/A

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

    18. /-lowering-/.f64 N/A

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

    19. pow1/3 N/A

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

    20. cbrt-lowering-cbrt.f64 96.2

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

12. Applied egg-rr 96.2%

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

Alternative 2: 89.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < -4.99999999999999955e-308

   1. Initial program 47.0%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. neg-lowering-neg.f64 26.5

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

   5. Simplified 26.5%

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

   6. Taylor expanded in g around inf

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

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

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

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

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

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

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

      4. cbrt-lowering-cbrt.f64 76.2

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

   8. Simplified 76.2%

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

      1. *-commutative N/A

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

      2. cbrt-unprod N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. cbrt-prod N/A

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

      8. pow1/3 N/A

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

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

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

      10. inv-pow N/A

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

      11. pow-pow N/A

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

      12. metadata-eval N/A

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

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

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

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

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

      15. cbrt-lowering-cbrt.f64 91.1

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

   10. Applied egg-rr 91.1%

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

   if -4.99999999999999955e-308 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 40.2%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. neg-lowering-neg.f64 25.9

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

   5. Simplified 25.9%

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

   6. Taylor expanded in g around inf

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

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

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

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

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

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

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

      4. cbrt-lowering-cbrt.f64 74.1

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

   8. Simplified 74.1%

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

      1. *-commutative N/A

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

      2. cbrt-unprod N/A

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

      3. neg-mul-1 N/A

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

      4. div-inv N/A

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

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

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. pow-lowering-pow.f64 88.9

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

   10. Applied egg-rr 88.9%

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

4. Final simplification 90.0%

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

Alternative 3: 81.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 40000000000000:\\ \;\;\;\;\sqrt[3]{\frac{g}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (/ 1.0 (* a 2.0)) 40000000000000.0)
   (cbrt (/ g (- a)))
   (* (cbrt (- g)) (pow a -0.3333333333333333))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= 40000000000000.0) {
		tmp = cbrt((g / -a));
	} else {
		tmp = cbrt(-g) * pow(a, -0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= 40000000000000.0) {
		tmp = Math.cbrt((g / -a));
	} else {
		tmp = Math.cbrt(-g) * Math.pow(a, -0.3333333333333333);
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(1.0 / Float64(a * 2.0)) <= 40000000000000.0)
		tmp = cbrt(Float64(g / Float64(-a)));
	else
		tmp = Float64(cbrt(Float64(-g)) * (a ^ -0.3333333333333333));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 40000000000000.0], N[Power[N[(g / (-a)), $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[(-g), 1/3], $MachinePrecision] * N[Power[a, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 4e13

   1. Initial program 44.9%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. neg-lowering-neg.f64 27.5

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

   5. Simplified 27.5%

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

   6. Taylor expanded in g around inf

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

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

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

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

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

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

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

      4. cbrt-lowering-cbrt.f64 82.1

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

   8. Simplified 82.1%

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

      1. *-commutative N/A

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

      2. cbrt-unprod N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-frac-neg2 N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. neg-lowering-neg.f64 82.1

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

   10. Applied egg-rr 82.1%

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

   if 4e13 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 39.7%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in g around inf

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. neg-lowering-neg.f64 22.4

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

   5. Simplified 22.4%

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

   6. Taylor expanded in g around inf

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

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

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

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

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

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

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

      4. cbrt-lowering-cbrt.f64 54.5

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

   8. Simplified 54.5%

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

      1. *-commutative N/A

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

      2. cbrt-unprod N/A

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

      3. neg-mul-1 N/A

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

      4. div-inv N/A

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

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

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. inv-pow N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. pow-lowering-pow.f64 88.4

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

   10. Applied egg-rr 88.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 40000000000000:\\ \;\;\;\;\sqrt[3]{\frac{g}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (g h a) :precision binary64 (/ (cbrt (- g)) (cbrt a)))

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(-g)) / cbrt(a))
end

Wolfram

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

Derivation

1. Initial program 43.6%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

   4. neg-lowering-neg.f64 26.2

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

5. Simplified 26.2%

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

6. Taylor expanded in g around inf

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

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

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

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

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

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

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

   4. cbrt-lowering-cbrt.f64 75.2

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

8. Simplified 75.2%

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

   1. *-commutative N/A

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

   2. cbrt-unprod N/A

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

   3. neg-mul-1 N/A

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

   4. distribute-frac-neg N/A

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

   5. cbrt-div N/A

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

   6. /-lowering-/.f64 N/A

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

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

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

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

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

   9. cbrt-lowering-cbrt.f64 96.0

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

10. Applied egg-rr 96.0%

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

Alternative 5: 74.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.6%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

   4. neg-lowering-neg.f64 26.2

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

5. Simplified 26.2%

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

6. Taylor expanded in g around inf

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

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

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

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

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

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

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

   4. cbrt-lowering-cbrt.f64 75.2

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

8. Simplified 75.2%

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

   1. *-commutative N/A

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

   2. cbrt-unprod N/A

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

   3. neg-mul-1 N/A

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

   4. distribute-frac-neg N/A

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

   5. cbrt-div N/A

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

   6. /-lowering-/.f64 N/A

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

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

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

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

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

   9. cbrt-lowering-cbrt.f64 96.0

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

10. Applied egg-rr 96.0%

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

    1. clear-num N/A

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

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

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

    3. cbrt-undiv N/A

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

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

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

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

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

    6. neg-lowering-neg.f64 75.3

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

12. Applied egg-rr 75.3%

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

Alternative 6: 74.0% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (g h a) :precision binary64 (cbrt (/ g (- a))))

C

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

Java

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

Julia

function code(g, h, a)
	return cbrt(Float64(g / Float64(-a)))
end

Wolfram

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

Derivation

1. Initial program 43.6%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

   4. neg-lowering-neg.f64 26.2

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

5. Simplified 26.2%

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

6. Taylor expanded in g around inf

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

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

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

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

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

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

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

   4. cbrt-lowering-cbrt.f64 75.2

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

8. Simplified 75.2%

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

   1. *-commutative N/A

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

   2. cbrt-unprod N/A

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

   3. neg-mul-1 N/A

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

   4. distribute-frac-neg2 N/A

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

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

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

   6. /-lowering-/.f64 N/A

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

   7. neg-lowering-neg.f64 75.2

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

10. Applied egg-rr 75.2%

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

Alternative 7: 1.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{a}{g}\right)}^{-0.3333333333333333} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (pow (/ a g) -0.3333333333333333))

C

double code(double g, double h, double a) {
	return pow((a / g), -0.3333333333333333);
}

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = (a / g) ** (-0.3333333333333333d0)
end function

Java

public static double code(double g, double h, double a) {
	return Math.pow((a / g), -0.3333333333333333);
}

Python

def code(g, h, a):
	return math.pow((a / g), -0.3333333333333333)

Julia

function code(g, h, a)
	return Float64(a / g) ^ -0.3333333333333333
end

MATLAB

function tmp = code(g, h, a)
	tmp = (a / g) ^ -0.3333333333333333;
end

Wolfram

code[g_, h_, a_] := N[Power[N[(a / g), $MachinePrecision], -0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 43.6%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

   4. neg-lowering-neg.f64 26.2

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

5. Simplified 26.2%

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

6. Taylor expanded in g around inf

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

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

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

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

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

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

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

   4. cbrt-lowering-cbrt.f64 75.2

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

8. Simplified 75.2%

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

   1. *-commutative N/A

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

   2. cbrt-unprod N/A

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

   3. neg-mul-1 N/A

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

   4. distribute-frac-neg2 N/A

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

   5. pow1/3 N/A

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

   6. sqr-pow N/A

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

   7. pow-prod-down N/A

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

   8. distribute-frac-neg2 N/A

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

   9. distribute-frac-neg2 N/A

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

   10. sqr-neg N/A

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

   11. pow-prod-down N/A

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

   12. sqr-pow N/A

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

   13. clear-num N/A

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

   14. inv-pow N/A

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

   15. pow-pow N/A

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

   16. metadata-eval N/A

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

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

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

   18. /-lowering-/.f64 1.4

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

10. Applied egg-rr 1.4%

    \[\leadsto \color{blue}{{\left(\frac{a}{g}\right)}^{-0.3333333333333333}} \]
11. Add Preprocessing

Alternative 8: 1.4% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (g h a) :precision binary64 (cbrt (/ g a)))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.6%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Add Preprocessing

3. Taylor expanded in g around inf

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

   4. neg-lowering-neg.f64 26.2

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

5. Simplified 26.2%

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

6. Taylor expanded in g around inf

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

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

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

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

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

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

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

   4. cbrt-lowering-cbrt.f64 75.2

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

8. Simplified 75.2%

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

   1. *-commutative N/A

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

   2. cbrt-unprod N/A

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

   3. neg-mul-1 N/A

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

   4. distribute-frac-neg2 N/A

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

   5. pow1/3 N/A

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

   6. sqr-pow N/A

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

   7. pow-prod-down N/A

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

   8. distribute-frac-neg2 N/A

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

   9. distribute-frac-neg2 N/A

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

   10. sqr-neg N/A

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

   11. pow-prod-down N/A

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

   12. sqr-pow N/A

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

   13. pow1/3 N/A

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

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

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

   15. /-lowering-/.f64 1.3

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

10. Applied egg-rr 1.3%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))