2-ancestry mixing, positive discriminant

Percentage Accurate: 43.9% → 95.7%

Time: 24.4s

Alternatives: 4

Speedup: 2.1×

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: 95.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g \cdot -2} + \sqrt[3]{\left(g - g\right) \cdot \frac{-0.5}{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (* (cbrt (/ 0.5 a)) (cbrt (* g -2.0))) (cbrt (* (- g g) (/ -0.5 a)))))

C

double code(double g, double h, double a) {
	return (cbrt((0.5 / a)) * cbrt((g * -2.0))) + cbrt(((g - g) * (-0.5 / a)));
}

Java

public static double code(double g, double h, double a) {
	return (Math.cbrt((0.5 / a)) * Math.cbrt((g * -2.0))) + Math.cbrt(((g - g) * (-0.5 / a)));
}

Julia

function code(g, h, a)
	return Float64(Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(g * -2.0))) + cbrt(Float64(Float64(g - g) * Float64(-0.5 / a))))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g * -2.0), $MachinePrecision], 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 40.7%

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

3. Simplified 40.7%

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

4. Taylor expanded in g around -inf 27.7%

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

   1. *-commutative 27.7%

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

6. Simplified 27.7%

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

7. Taylor expanded in g around -inf 74.5%

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

   1. neg-mul-1 74.5%

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

9. Simplified 74.5%

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

    1. cbrt-prod 96.2%

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

11. Applied egg-rr 96.2%

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

12. Final simplification 96.2%

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

Alternative 2: 95.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{0} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (+ (cbrt 0.0) (/ (cbrt (- g)) (cbrt a))))

C

double code(double g, double h, double a) {
	return cbrt(0.0) + (cbrt(-g) / cbrt(a));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(0.0) + (Math.cbrt(-g) / Math.cbrt(a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(0.0) + Float64(cbrt(Float64(-g)) / cbrt(a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[0.0, 1/3], $MachinePrecision] + N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

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

3. Simplified 40.7%

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

4. Taylor expanded in g around -inf 27.7%

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

   1. *-commutative 27.7%

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

6. Simplified 27.7%

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

7. Taylor expanded in g around inf 15.4%

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

   1. add-log-exp 27.4%

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

   2. *-commutative 27.4%

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

   3. exp-prod 18.3%

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

   4. add-sqr-sqrt 12.5%

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

   5. sqrt-unprod 16.8%

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

   6. *-commutative 16.8%

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

   7. *-commutative 16.8%

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

   8. swap-sqr 16.8%

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

   9. metadata-eval 16.8%

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

   10. metadata-eval 16.8%

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

   11. swap-sqr 16.8%

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

   12. count-2 16.8%

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

   13. count-2 16.8%

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

   14. sqrt-unprod 4.3%

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

   15. add-sqr-sqrt 15.6%

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

   16. add-sqr-sqrt 14.7%

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

   17. sqrt-unprod 38.1%

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

   18. frac-times 40.4%

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

   19. metadata-eval 40.4%

       \[\leadsto \sqrt[3]{\log \left({\left(e^{g + g}\right)}^{\left(\sqrt{\frac{\color{blue}{0.25}}{a \cdot a}}\right)}\right)} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   20. metadata-eval 40.4%

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

   21. frac-times 38.1%

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

   22. sqrt-unprod 16.3%

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

   23. add-sqr-sqrt 18.3%

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

9. Applied egg-rr 74.5%

   \[\leadsto \sqrt[3]{\color{blue}{0}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 74.5%

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

    2. cbrt-div 96.0%

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

    3. count-2 96.0%

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

11. Applied egg-rr 96.0%

    \[\leadsto \sqrt[3]{0} + \color{blue}{\frac{\sqrt[3]{\left(2 \cdot g\right) \cdot -0.5}}{\sqrt[3]{a}}} \]
12. Step-by-step derivation

    1. *-commutative 96.0%

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

    2. associate-*r* 96.0%

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

    3. metadata-eval 96.0%

       \[\leadsto \sqrt[3]{0} + \frac{\sqrt[3]{\color{blue}{-1} \cdot g}}{\sqrt[3]{a}} \]

    4. mul-1-neg 96.0%

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

13. Simplified 96.0%

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

14. Final simplification 96.0%

    \[\leadsto \sqrt[3]{0} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \]

Alternative 3: 74.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{0} + \sqrt[3]{\frac{-g}{a}} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (+ (cbrt 0.0) (cbrt (/ (- g) a))))

C

double code(double g, double h, double a) {
	return cbrt(0.0) + cbrt((-g / a));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(0.0) + Math.cbrt((-g / a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(0.0) + cbrt(Float64(Float64(-g) / a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[0.0, 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

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

3. Simplified 40.7%

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

4. Taylor expanded in g around -inf 27.7%

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

   1. *-commutative 27.7%

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

6. Simplified 27.7%

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

7. Taylor expanded in g around inf 15.4%

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

   1. add-log-exp 27.4%

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

   2. *-commutative 27.4%

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

   3. exp-prod 18.3%

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

   4. add-sqr-sqrt 12.5%

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

   5. sqrt-unprod 16.8%

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

   6. *-commutative 16.8%

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

   7. *-commutative 16.8%

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

   8. swap-sqr 16.8%

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

   9. metadata-eval 16.8%

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

   10. metadata-eval 16.8%

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

   11. swap-sqr 16.8%

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

   12. count-2 16.8%

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

   13. count-2 16.8%

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

   14. sqrt-unprod 4.3%

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

   15. add-sqr-sqrt 15.6%

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

   16. add-sqr-sqrt 14.7%

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

   17. sqrt-unprod 38.1%

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

   18. frac-times 40.4%

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

   19. metadata-eval 40.4%

       \[\leadsto \sqrt[3]{\log \left({\left(e^{g + g}\right)}^{\left(\sqrt{\frac{\color{blue}{0.25}}{a \cdot a}}\right)}\right)} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   20. metadata-eval 40.4%

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

   21. frac-times 38.1%

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

   22. sqrt-unprod 16.3%

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

   23. add-sqr-sqrt 18.3%

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

9. Applied egg-rr 74.5%

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

10. Taylor expanded in g around 0 74.5%

    \[\leadsto \sqrt[3]{0} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
11. Step-by-step derivation

    1. associate-*r/ 74.5%

       \[\leadsto \sqrt[3]{0} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

    2. mul-1-neg 74.5%

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

12. Simplified 74.5%

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

13. Final simplification 74.5%

    \[\leadsto \sqrt[3]{0} + \sqrt[3]{\frac{-g}{a}} \]

Alternative 4: 3.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{0} + \sqrt[3]{0} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (+ (cbrt 0.0) (cbrt 0.0)))

C

double code(double g, double h, double a) {
	return cbrt(0.0) + cbrt(0.0);
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(0.0) + Math.cbrt(0.0);
}

Julia

function code(g, h, a)
	return Float64(cbrt(0.0) + cbrt(0.0))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[0.0, 1/3], $MachinePrecision] + N[Power[0.0, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

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

3. Simplified 40.7%

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

4. Taylor expanded in g around -inf 27.7%

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

   1. *-commutative 27.7%

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

6. Simplified 27.7%

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

7. Taylor expanded in g around inf 15.4%

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

   1. add-log-exp 27.4%

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

   2. *-commutative 27.4%

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

   3. exp-prod 18.3%

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

   4. add-sqr-sqrt 12.5%

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

   5. sqrt-unprod 16.8%

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

   6. *-commutative 16.8%

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

   7. *-commutative 16.8%

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

   8. swap-sqr 16.8%

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

   9. metadata-eval 16.8%

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

   10. metadata-eval 16.8%

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

   11. swap-sqr 16.8%

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

   12. count-2 16.8%

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

   13. count-2 16.8%

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

   14. sqrt-unprod 4.3%

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

   15. add-sqr-sqrt 15.6%

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

   16. add-sqr-sqrt 14.7%

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

   17. sqrt-unprod 38.1%

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

   18. frac-times 40.4%

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

   19. metadata-eval 40.4%

       \[\leadsto \sqrt[3]{\log \left({\left(e^{g + g}\right)}^{\left(\sqrt{\frac{\color{blue}{0.25}}{a \cdot a}}\right)}\right)} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]

   20. metadata-eval 40.4%

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

   21. frac-times 38.1%

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

   22. sqrt-unprod 16.3%

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

   23. add-sqr-sqrt 18.3%

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

9. Applied egg-rr 74.5%

   \[\leadsto \sqrt[3]{\color{blue}{0}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \]
10. Step-by-step derivation

    1. *-commutative 74.5%

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

    2. clear-num 74.5%

       \[\leadsto \sqrt[3]{0} + \sqrt[3]{\color{blue}{\frac{1}{\frac{a}{-0.5}}} \cdot \left(g + g\right)} \]

    3. flip-+ 0.0%

       \[\leadsto \sqrt[3]{0} + \sqrt[3]{\frac{1}{\frac{a}{-0.5}} \cdot \color{blue}{\frac{g \cdot g - g \cdot g}{g - g}}} \]

    4. frac-times 0.0%

       \[\leadsto \sqrt[3]{0} + \sqrt[3]{\color{blue}{\frac{1 \cdot \left(g \cdot g - g \cdot g\right)}{\frac{a}{-0.5} \cdot \left(g - g\right)}}} \]

    5. *-un-lft-identity 0.0%

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

    6. pow2 0.0%

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

    7. pow2 0.0%

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

    8. *-un-lft-identity 0.0%

       \[\leadsto \sqrt[3]{0} + \sqrt[3]{\frac{{g}^{2} - {g}^{2}}{\frac{a}{-0.5} \cdot \left(\color{blue}{1 \cdot g} - g\right)}} \]

    9. fma-neg 0.0%

       \[\leadsto \sqrt[3]{0} + \sqrt[3]{\frac{{g}^{2} - {g}^{2}}{\frac{a}{-0.5} \cdot \color{blue}{\mathsf{fma}\left(1, g, -g\right)}}} \]

    10. add-sqr-sqrt 0.4%

        \[\leadsto \sqrt[3]{0} + \sqrt[3]{\frac{{g}^{2} - {g}^{2}}{\frac{a}{-0.5} \cdot \mathsf{fma}\left(1, g, \color{blue}{\sqrt{-g} \cdot \sqrt{-g}}\right)}} \]

    11. sqrt-unprod 0.6%

        \[\leadsto \sqrt[3]{0} + \sqrt[3]{\frac{{g}^{2} - {g}^{2}}{\frac{a}{-0.5} \cdot \mathsf{fma}\left(1, g, \color{blue}{\sqrt{\left(-g\right) \cdot \left(-g\right)}}\right)}} \]

    12. sqr-neg 0.6%

        \[\leadsto \sqrt[3]{0} + \sqrt[3]{\frac{{g}^{2} - {g}^{2}}{\frac{a}{-0.5} \cdot \mathsf{fma}\left(1, g, \sqrt{\color{blue}{g \cdot g}}\right)}} \]

    13. sqrt-unprod 0.6%

        \[\leadsto \sqrt[3]{0} + \sqrt[3]{\frac{{g}^{2} - {g}^{2}}{\frac{a}{-0.5} \cdot \mathsf{fma}\left(1, g, \color{blue}{\sqrt{g} \cdot \sqrt{g}}\right)}} \]

    14. add-sqr-sqrt 1.4%

        \[\leadsto \sqrt[3]{0} + \sqrt[3]{\frac{{g}^{2} - {g}^{2}}{\frac{a}{-0.5} \cdot \mathsf{fma}\left(1, g, \color{blue}{g}\right)}} \]

    15. fma-def 1.4%

        \[\leadsto \sqrt[3]{0} + \sqrt[3]{\frac{{g}^{2} - {g}^{2}}{\frac{a}{-0.5} \cdot \color{blue}{\left(1 \cdot g + g\right)}}} \]

    16. *-un-lft-identity 1.4%

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

    17. count-2 1.4%

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

11. Applied egg-rr 1.4%

    \[\leadsto \sqrt[3]{0} + \sqrt[3]{\color{blue}{\frac{{g}^{2} - {g}^{2}}{\frac{a}{-0.5} \cdot \left(2 \cdot g\right)}}} \]
12. Step-by-step derivation

    1. div-sub 1.3%

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

    2. +-inverses 2.9%

       \[\leadsto \sqrt[3]{0} + \sqrt[3]{\color{blue}{0}} \]

13. Simplified 2.9%

    \[\leadsto \sqrt[3]{0} + \sqrt[3]{\color{blue}{0}} \]

14. Final simplification 2.9%

    \[\leadsto \sqrt[3]{0} + \sqrt[3]{0} \]

Reproduce

herbie shell --seed 2023326 
(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))))))))