2-ancestry mixing, positive discriminant

Percentage Accurate: 43.9% → 95.6%

Time: 34.8s

Alternatives: 7

Speedup: 2.0×

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: 43.9% 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

\[\begin{array}{l} \\ \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

3. Simplified 46.5%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
6. Step-by-step derivation

   1. *-commutative 28.1%

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

7. Simplified 28.1%

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

8. Taylor expanded in g around -inf 76.5%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{-1 \cdot g}\right) \cdot \frac{-0.5}{a}} \]
9. Step-by-step derivation

   1. neg-mul-1 76.5%

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

10. Simplified 76.5%

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{\left(-g\right)}\right) \cdot \frac{-0.5}{a}} \]
11. Step-by-step derivation

    1. associate-*l/ 76.5%

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

    2. cbrt-div 95.8%

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

    3. *-commutative 95.8%

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

    4. associate-*r* 96.2%

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

    5. metadata-eval 96.2%

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

    6. neg-mul-1 96.2%

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

12. Applied egg-rr 96.2%

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

13. Final simplification 96.2%

    \[\leadsto \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} \]
14. Add Preprocessing

Alternative 2: 89.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;a \leq -6.2 \cdot 10^{-69}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-2}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{1}{\frac{a}{-g}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (cbrt (* (- g g) (/ -0.5 a)))))
   (if (<= a -6.2e-69)
     (+ t_0 (cbrt (* (/ 0.5 a) (* g -2.0))))
     (if (<= a 3.9e-36)
       (+ (/ (cbrt (- g)) (cbrt a)) (cbrt -2.0))
       (+ t_0 (cbrt (/ 1.0 (/ a (- g)))))))))

C

double code(double g, double h, double a) {
	double t_0 = cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -6.2e-69) {
		tmp = t_0 + cbrt(((0.5 / a) * (g * -2.0)));
	} else if (a <= 3.9e-36) {
		tmp = (cbrt(-g) / cbrt(a)) + cbrt(-2.0);
	} else {
		tmp = t_0 + cbrt((1.0 / (a / -g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double t_0 = Math.cbrt(((g - g) * (-0.5 / a)));
	double tmp;
	if (a <= -6.2e-69) {
		tmp = t_0 + Math.cbrt(((0.5 / a) * (g * -2.0)));
	} else if (a <= 3.9e-36) {
		tmp = (Math.cbrt(-g) / Math.cbrt(a)) + Math.cbrt(-2.0);
	} else {
		tmp = t_0 + Math.cbrt((1.0 / (a / -g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	t_0 = cbrt(Float64(Float64(g - g) * Float64(-0.5 / a)))
	tmp = 0.0
	if (a <= -6.2e-69)
		tmp = Float64(t_0 + cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))));
	elseif (a <= 3.9e-36)
		tmp = Float64(Float64(cbrt(Float64(-g)) / cbrt(a)) + cbrt(-2.0));
	else
		tmp = Float64(t_0 + cbrt(Float64(1.0 / Float64(a / Float64(-g)))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[Power[N[(N[(g - g), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[a, -6.2e-69], N[(t$95$0 + N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.9e-36], N[(N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[-2.0, 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[N[(1.0 / N[(a / (-g)), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.1999999999999999e-69

   1. Initial program 45.8%

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

   3. Simplified 45.8%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in g around -inf 24.2%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 24.2%

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

   7. Simplified 24.2%

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

   8. Taylor expanded in g around -inf 89.6%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{-1 \cdot g}\right) \cdot \frac{-0.5}{a}} \]
   9. Step-by-step derivation

      1. neg-mul-1 89.6%

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

   10. Simplified 89.6%

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

   if -6.1999999999999999e-69 < a < 3.9000000000000001e-36

   1. Initial program 41.0%

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

   3. Simplified 41.0%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in g around -inf 27.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 27.9%

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

   7. Simplified 27.9%

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

   8. Taylor expanded in g around inf 11.5%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{g}\right) \cdot \frac{-0.5}{a}} \]
   9. Step-by-step derivation

      1. associate-*l/ 11.5%

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

      2. associate-/l* 11.5%

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

      3. add-sqr-sqrt 6.5%

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

      4. sqrt-unprod 11.1%

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

      5. *-commutative 11.1%

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

      6. *-commutative 11.1%

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

      7. swap-sqr 11.1%

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

      8. metadata-eval 11.1%

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

      9. metadata-eval 11.1%

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

      10. swap-sqr 11.1%

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

      11. count-2 11.1%

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

      12. count-2 11.1%

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

      13. sqrt-unprod 0.9%

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

      14. add-sqr-sqrt 1.9%

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

      15. flip-+ 0.0%

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

      16. +-inverses 0.0%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{\color{blue}{0}}{g - g}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

      17. +-inverses 0.0%

          \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{0}{\color{blue}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   10. Applied egg-rr 0.0%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{\frac{a}{\frac{0}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   12. Simplified 44.7%

       \[\leadsto \sqrt[3]{\color{blue}{-2}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 18.2%

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

       2. sqrt-unprod 6.4%

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

       3. swap-sqr 3.0%

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

       4. count-2 3.0%

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

       5. count-2 3.0%

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

       6. swap-sqr 3.0%

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

       7. metadata-eval 3.0%

          \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\sqrt{\left(\color{blue}{4} \cdot \left(g \cdot g\right)\right) \cdot \left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right)}} \]

       8. metadata-eval 3.0%

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

       9. swap-sqr 3.0%

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

       10. *-commutative 3.0%

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

       11. *-commutative 3.0%

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

       12. frac-times 3.0%

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

       13. metadata-eval 3.0%

           \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\sqrt{\left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right) \cdot \frac{\color{blue}{0.25}}{a \cdot a}}} \]

       14. metadata-eval 3.0%

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

       15. frac-times 3.0%

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

       16. swap-sqr 6.4%

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

       17. *-commutative 6.4%

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

       18. *-commutative 6.4%

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

       19. sqrt-unprod 18.2%

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

       20. add-sqr-sqrt 44.7%

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

   14. Applied egg-rr 91.6%

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

   if 3.9000000000000001e-36 < a

   1. Initial program 52.7%

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

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in g around -inf 32.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 32.8%

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

   7. Simplified 32.8%

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

   8. Taylor expanded in g around -inf 89.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{-1 \cdot g}\right) \cdot \frac{-0.5}{a}} \]
   9. Step-by-step derivation

      1. neg-mul-1 89.9%

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

   10. Simplified 89.9%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{\left(-g\right)}\right) \cdot \frac{-0.5}{a}} \]
   11. Step-by-step derivation

       1. associate-*l/ 90.0%

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

       2. clear-num 90.0%

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

       3. *-commutative 90.0%

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

       4. associate-*r* 91.2%

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

       5. metadata-eval 91.2%

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

       6. neg-mul-1 91.2%

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

   12. Applied egg-rr 91.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{-69}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)}\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt[3]{-g}}{\sqrt[3]{a}} + \sqrt[3]{-2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{1}{\frac{a}{-g}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;g \leq -1.8 \cdot 10^{+19} \lor \neg \left(g \leq 1350000000\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \frac{-2}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (or (<= g -1.8e+19) (not (<= g 1350000000.0)))
   (+ (cbrt (* (/ 0.5 a) (* g -2.0))) (/ -2.0 (cbrt a)))
   (+ (cbrt (- g)) (cbrt (* (/ -0.5 a) (+ g g))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((g <= -1.8e+19) || !(g <= 1350000000.0)) {
		tmp = cbrt(((0.5 / a) * (g * -2.0))) + (-2.0 / cbrt(a));
	} else {
		tmp = cbrt(-g) + cbrt(((-0.5 / a) * (g + g)));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((g <= -1.8e+19) || !(g <= 1350000000.0)) {
		tmp = Math.cbrt(((0.5 / a) * (g * -2.0))) + (-2.0 / Math.cbrt(a));
	} else {
		tmp = Math.cbrt(-g) + Math.cbrt(((-0.5 / a) * (g + g)));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if ((g <= -1.8e+19) || !(g <= 1350000000.0))
		tmp = Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))) + Float64(-2.0 / cbrt(a)));
	else
		tmp = Float64(cbrt(Float64(-g)) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + g))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[Or[LessEqual[g, -1.8e+19], N[Not[LessEqual[g, 1350000000.0]], $MachinePrecision]], N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(-2.0 / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[(-g), 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if g < -1.8e19 or 1.35e9 < g

   1. Initial program 37.6%

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

   3. Simplified 37.6%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in g around -inf 21.3%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 21.3%

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

   7. Simplified 21.3%

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

   8. Taylor expanded in g around inf 15.3%

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

   10. Applied egg-rr 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \color{blue}{\frac{\frac{0}{0}}{\sqrt[3]{a}}} \]

   12. Simplified 71.5%

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

   if -1.8e19 < g < 1.35e9

   1. Initial program 69.0%

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

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in g around -inf 45.7%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 45.7%

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

   7. Simplified 45.7%

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

   8. Taylor expanded in g around inf 16.6%

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

   9. Taylor expanded in a around 0 16.6%

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

   11. Simplified 38.8%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.8 \cdot 10^{+19} \lor \neg \left(g \leq 1350000000\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \frac{-2}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-g}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

3. Simplified 46.5%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
6. Step-by-step derivation

   1. *-commutative 28.1%

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

7. Simplified 28.1%

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

8. Taylor expanded in g around -inf 76.5%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{-1 \cdot g}\right) \cdot \frac{-0.5}{a}} \]
9. Step-by-step derivation

   1. neg-mul-1 76.5%

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

10. Simplified 76.5%

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{\left(-g\right)}\right) \cdot \frac{-0.5}{a}} \]
11. Step-by-step derivation

    1. associate-*l/ 76.5%

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

    2. *-commutative 76.5%

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

    3. associate-*r* 76.8%

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

    4. metadata-eval 76.8%

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

    5. neg-mul-1 76.8%

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

12. Applied egg-rr 76.8%

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

13. Final simplification 76.8%

    \[\leadsto \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-g}{a}} \]
14. Add Preprocessing

Alternative 5: 43.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

3. Simplified 46.5%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
6. Step-by-step derivation

   1. *-commutative 28.1%

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

7. Simplified 28.1%

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

8. Taylor expanded in g around inf 15.7%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{g}\right) \cdot \frac{-0.5}{a}} \]
9. Step-by-step derivation

   1. associate-*l/ 15.7%

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

   2. associate-/l* 15.7%

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

   3. add-sqr-sqrt 7.9%

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

   4. sqrt-unprod 9.9%

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

   5. *-commutative 9.9%

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

   6. *-commutative 9.9%

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

   7. swap-sqr 9.9%

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

   8. metadata-eval 9.9%

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

   9. metadata-eval 9.9%

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

   10. swap-sqr 9.9%

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

   11. count-2 9.9%

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

   12. count-2 9.9%

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

   13. sqrt-unprod 1.8%

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

   14. add-sqr-sqrt 3.3%

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

   15. flip-+ 0.0%

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

   16. +-inverses 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{\color{blue}{0}}{g - g}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   17. +-inverses 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{0}{\color{blue}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{\frac{a}{\frac{0}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 40.3%

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

13. Taylor expanded in g around 0 40.6%

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

    1. neg-mul-1 40.6%

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

    2. distribute-neg-frac 40.6%

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

15. Simplified 40.6%

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

16. Final simplification 40.6%

    \[\leadsto \sqrt[3]{-2} + \sqrt[3]{\frac{-g}{a}} \]
17. Add Preprocessing

Alternative 6: 4.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

3. Simplified 46.5%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
6. Step-by-step derivation

   1. *-commutative 28.1%

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

7. Simplified 28.1%

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

8. Taylor expanded in g around inf 15.7%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{g}\right) \cdot \frac{-0.5}{a}} \]
9. Step-by-step derivation

   1. associate-*l/ 15.7%

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

   2. associate-/l* 15.7%

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

   3. add-sqr-sqrt 7.9%

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

   4. sqrt-unprod 9.9%

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

   5. *-commutative 9.9%

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

   6. *-commutative 9.9%

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

   7. swap-sqr 9.9%

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

   8. metadata-eval 9.9%

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

   9. metadata-eval 9.9%

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

   10. swap-sqr 9.9%

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

   11. count-2 9.9%

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

   12. count-2 9.9%

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

   13. sqrt-unprod 1.8%

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

   14. add-sqr-sqrt 3.3%

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

   15. flip-+ 0.0%

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

   16. +-inverses 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{\color{blue}{0}}{g - g}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   17. +-inverses 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{0}{\color{blue}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{\frac{a}{\frac{0}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 40.3%

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

14. Applied egg-rr 0.0%

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

16. Simplified 4.9%

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

17. Final simplification 4.9%

    \[\leadsto \sqrt[3]{-2} + \frac{-2}{\sqrt[3]{a}} \]
18. Add Preprocessing

Alternative 7: 4.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{-2} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[g_, h_, a_] := N[Power[-2.0, 1/3], $MachinePrecision]

Derivation

1. Initial program 46.5%

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

3. Simplified 46.5%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
4. Add Preprocessing

5. Taylor expanded in g around -inf 28.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
6. Step-by-step derivation

   1. *-commutative 28.1%

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

7. Simplified 28.1%

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

8. Taylor expanded in g around inf 15.7%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\left(g + \color{blue}{g}\right) \cdot \frac{-0.5}{a}} \]
9. Step-by-step derivation

   1. associate-*l/ 15.7%

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

   2. associate-/l* 15.7%

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

   3. add-sqr-sqrt 7.9%

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

   4. sqrt-unprod 9.9%

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

   5. *-commutative 9.9%

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

   6. *-commutative 9.9%

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

   7. swap-sqr 9.9%

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

   8. metadata-eval 9.9%

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

   9. metadata-eval 9.9%

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

   10. swap-sqr 9.9%

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

   11. count-2 9.9%

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

   12. count-2 9.9%

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

   13. sqrt-unprod 1.8%

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

   14. add-sqr-sqrt 3.3%

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

   15. flip-+ 0.0%

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

   16. +-inverses 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{\color{blue}{0}}{g - g}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   17. +-inverses 0.0%

       \[\leadsto \sqrt[3]{\frac{0.5}{\frac{a}{\frac{0}{\color{blue}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

10. Applied egg-rr 0.0%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5}{\frac{a}{\frac{0}{0}}}}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

12. Simplified 40.3%

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

13. Taylor expanded in g around 0 4.7%

    \[\leadsto \color{blue}{\sqrt[3]{-2}} \]

14. Final simplification 4.7%

    \[\leadsto \sqrt[3]{-2} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))))))