2-ancestry mixing, positive discriminant

Percentage Accurate: 43.7% → 95.6%

Time: 16.0s

Alternatives: 7

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: 43.7% 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: 95.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.4%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. lower-/.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. lower-neg.f64 29.5

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

5. Simplified 29.5%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 78.6

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

8. Simplified 78.6%

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

   1. lift-/.f64 N/A

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

   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. lift-/.f64 N/A

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

   5. distribute-frac-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. cbrt-div N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. frac-2neg N/A

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

   11. div-inv N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-neg.f64 96.8

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

10. Applied egg-rr 96.8%

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

    1. lift-neg.f64 N/A

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

    2. lift-cbrt.f64 N/A

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

    3. lift-neg.f64 N/A

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

    4. lift-cbrt.f64 N/A

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

    5. lift-neg.f64 N/A

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

    6. un-div-inv N/A

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

    7. clear-num N/A

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

    8. metadata-eval N/A

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

    9. lift-neg.f64 N/A

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

    10. lift-neg.f64 N/A

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

    11. frac-2neg N/A

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

    12. lift-cbrt.f64 N/A

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

    13. lift-cbrt.f64 N/A

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

    14. cbrt-div N/A

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

    15. lift-neg.f64 N/A

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

    16. distribute-neg-frac2 N/A

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

    17. lift-/.f64 N/A

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

    18. neg-mul-1 N/A

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

    19. metadata-eval N/A

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

    20. lift-/.f64 N/A

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

    21. times-frac N/A

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

    22. lift-*.f64 N/A

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

    23. *-commutative N/A

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

    24. lift-*.f64 N/A

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

    25. *-lft-identity N/A

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

12. Applied egg-rr 96.8%

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

Alternative 2: 81.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 5 \cdot 10^{+37}:\\ \;\;\;\;-\sqrt[3]{\frac{\frac{1}{a}}{\frac{1}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (/ 1.0 (* a 2.0)) 5e+37)
   (- (cbrt (/ (/ 1.0 a) (/ 1.0 g))))
   (* (cbrt (- g)) (pow a -0.3333333333333333))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (a * 2.0)) <= 5e+37) {
		tmp = -cbrt(((1.0 / a) / (1.0 / g)));
	} 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+37) {
		tmp = -Math.cbrt(((1.0 / a) / (1.0 / g)));
	} 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+37)
		tmp = Float64(-cbrt(Float64(Float64(1.0 / a) / Float64(1.0 / g))));
	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+37], (-N[Power[N[(N[(1.0 / a), $MachinePrecision] / N[(1.0 / g), $MachinePrecision]), $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)) < 4.99999999999999989e37

   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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 25.6

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot \color{blue}{\left(h \cdot -0.5\right)}}{g}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   5. Simplified 25.6%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\frac{h \cdot \left(h \cdot -0.5\right)}{g}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   6. Taylor expanded in g around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 76.9

         \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot \left(h \cdot -0.5\right)}{g}} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

   8. Simplified 76.9%

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot \left(h \cdot -0.5\right)}{g}} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

   9. Taylor expanded in g around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-cbrt.f64 N/A

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

       4. lower-/.f64 82.2

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

   11. Simplified 82.2%

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

       1. clear-num N/A

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

       2. div-inv N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 82.3

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

   13. Applied egg-rr 82.3%

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

   if 4.99999999999999989e37 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 47.5%

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   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. lower-/.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. lower-neg.f64 34.1

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

   5. Simplified 34.1%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lower-/.f64 64.5

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

   8. Simplified 64.5%

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

      1. lift-/.f64 N/A

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

      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. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. cbrt-prod N/A

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

      11. lift-cbrt.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. lower-*.f64 94.1

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

      14. lift-cbrt.f64 N/A

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

      15. pow1/3 N/A

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

      16. lift-/.f64 N/A

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

      17. inv-pow N/A

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

      18. pow-pow N/A

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

      19. lower-pow.f64 N/A

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

      20. metadata-eval 87.7

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

   10. Applied egg-rr 87.7%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq 5 \cdot 10^{+37}:\\ \;\;\;\;-\sqrt[3]{\frac{\frac{1}{a}}{\frac{1}{g}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} \cdot {a}^{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.4%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. lower-/.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. lower-neg.f64 29.5

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

5. Simplified 29.5%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 78.6

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

8. Simplified 78.6%

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

   1. lift-/.f64 N/A

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

   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. lift-/.f64 N/A

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

   5. distribute-frac-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. cbrt-div N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. frac-2neg N/A

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

   11. div-inv N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-neg.f64 96.8

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

10. Applied egg-rr 96.8%

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

11. Taylor expanded in g around -inf

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

    1. lower-cbrt.f64 96.8

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

13. Simplified 96.8%

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

14. Final simplification 96.8%

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

Alternative 4: 95.7% 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 44.4%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. lower-/.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. lower-neg.f64 29.5

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

5. Simplified 29.5%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 78.6

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

8. Simplified 78.6%

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

   1. lift-/.f64 N/A

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

   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. lift-/.f64 N/A

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

   5. distribute-frac-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. cbrt-div N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. lift-/.f64 96.8

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

10. Applied egg-rr 96.8%

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

Alternative 5: 73.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.4%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 27.1

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot \color{blue}{\left(h \cdot -0.5\right)}}{g}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

5. Simplified 27.1%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\frac{h \cdot \left(h \cdot -0.5\right)}{g}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 74.4

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot \left(h \cdot -0.5\right)}{g}} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

8. Simplified 74.4%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot \left(h \cdot -0.5\right)}{g}} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

9. Taylor expanded in g around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 N/A

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

    3. lower-cbrt.f64 N/A

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

    4. lower-/.f64 78.6

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

11. Simplified 78.6%

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

    1. clear-num N/A

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

    2. div-inv N/A

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

    3. associate-/r* N/A

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

    4. lower-/.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. lower-/.f64 78.7

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

13. Applied egg-rr 78.7%

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

Alternative 6: 73.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.4%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 27.1

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot \color{blue}{\left(h \cdot -0.5\right)}}{g}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

5. Simplified 27.1%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\frac{h \cdot \left(h \cdot -0.5\right)}{g}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 74.4

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot \left(h \cdot -0.5\right)}{g}} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

8. Simplified 74.4%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot \left(h \cdot -0.5\right)}{g}} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

9. Taylor expanded in g around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 N/A

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

    3. lower-cbrt.f64 N/A

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

    4. lower-/.f64 78.6

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

11. Simplified 78.6%

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

    1. clear-num N/A

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

    2. associate-/r/ N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 78.7

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

13. Applied egg-rr 78.7%

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

14. Final simplification 78.7%

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

Alternative 7: 73.9% 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 Float64(-cbrt(Float64(g / a)))
end

Wolfram

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

Derivation

1. Initial program 44.4%

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

3. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 27.1

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot \color{blue}{\left(h \cdot -0.5\right)}}{g}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

5. Simplified 27.1%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\frac{h \cdot \left(h \cdot -0.5\right)}{g}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

6. Taylor expanded in g around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 74.4

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot \left(h \cdot -0.5\right)}{g}} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

8. Simplified 74.4%

   \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot \left(h \cdot -0.5\right)}{g}} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

9. Taylor expanded in g around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 N/A

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

    3. lower-cbrt.f64 N/A

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

    4. lower-/.f64 78.6

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

11. Simplified 78.6%

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

Reproduce

herbie shell --seed 2024211 
(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))))))))