2-ancestry mixing, positive discriminant

Percentage Accurate: 44.4% → 95.8%

Time: 23.3s

Alternatives: 3

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: 44.4% 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: 95.8% 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 47.0%

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

3. Simplified 47.0%

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

5. Taylor expanded in g around -inf 29.0%

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

   1. *-commutative 29.0%

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

7. Simplified 29.0%

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

8. Taylor expanded in g around -inf 69.8%

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

   1. neg-mul-1 69.8%

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

10. Simplified 69.8%

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

    1. associate-*l/ 69.8%

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

    2. cbrt-div 95.6%

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

    3. *-commutative 95.6%

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

    4. associate-*r* 95.6%

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

    5. metadata-eval 95.6%

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

    6. neg-mul-1 95.6%

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

12. Applied egg-rr 95.6%

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

13. Final simplification 95.6%

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

Alternative 2: 73.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{0} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt 0.0) (cbrt (* (/ -0.5 a) (+ g g)))))

C

double code(double g, double h, double a) {
	return cbrt(0.0) + cbrt(((-0.5 / a) * (g + g)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(0.0) + Math.cbrt(((-0.5 / a) * (g + g)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(0.0) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + g))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[0.0, 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.0%

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

3. Simplified 47.0%

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

5. Taylor expanded in g around -inf 29.0%

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

   1. *-commutative 29.0%

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

7. Simplified 29.0%

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

8. Taylor expanded in g around inf 14.7%

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

   1. add-log-exp 28.2%

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

   2. exp-prod 46.3%

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

   3. add-sqr-sqrt 45.3%

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

   4. sqrt-unprod 46.4%

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

   5. *-commutative 46.4%

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

   6. *-commutative 46.4%

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

   7. swap-sqr 46.4%

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

   8. metadata-eval 46.4%

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

   9. metadata-eval 46.4%

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

   10. swap-sqr 46.4%

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

   11. count-2 46.4%

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

   12. count-2 46.4%

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

   13. sqrt-unprod 43.7%

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

   14. add-sqr-sqrt 44.6%

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

   15. add-sqr-sqrt 43.7%

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

   16. sqrt-prod 54.1%

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

   17. pow2 54.1%

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

   18. pow2 54.1%

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

   19. sqr-neg 54.1%

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

   20. sqrt-unprod 48.9%

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

   21. add-sqr-sqrt 69.8%

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

   22. sub-neg 69.8%

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

   23. +-inverses 69.8%

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

   24. metadata-eval 69.8%

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

10. Applied egg-rr 69.8%

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

11. Final simplification 69.8%

    \[\leadsto \sqrt[3]{0} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)} \]
12. Add Preprocessing

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 47.0%

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

3. Simplified 47.0%

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

5. Taylor expanded in g around -inf 29.0%

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

   1. *-commutative 29.0%

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

7. Simplified 29.0%

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

8. Taylor expanded in g around inf 14.7%

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

   1. add-log-exp 28.2%

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

   2. exp-prod 46.3%

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

   3. add-sqr-sqrt 45.3%

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

   4. sqrt-unprod 46.4%

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

   5. *-commutative 46.4%

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

   6. *-commutative 46.4%

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

   7. swap-sqr 46.4%

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

   8. metadata-eval 46.4%

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

   9. metadata-eval 46.4%

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

   10. swap-sqr 46.4%

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

   11. count-2 46.4%

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

   12. count-2 46.4%

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

   13. sqrt-unprod 43.7%

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

   14. add-sqr-sqrt 44.6%

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

   15. add-sqr-sqrt 43.7%

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

   16. sqrt-prod 54.1%

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

   17. pow2 54.1%

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

   18. pow2 54.1%

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

   19. sqr-neg 54.1%

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

   20. sqrt-unprod 48.9%

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

   21. add-sqr-sqrt 69.8%

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

   22. sub-neg 69.8%

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

   23. +-inverses 69.8%

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

   24. metadata-eval 69.8%

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

10. Applied egg-rr 69.8%

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

11. Taylor expanded in g around 0 69.8%

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

    1. associate-*r/ 69.8%

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

    2. mul-1-neg 69.8%

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

13. Simplified 69.8%

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

14. Final simplification 69.8%

    \[\leadsto \sqrt[3]{0} + \sqrt[3]{\frac{-g}{a}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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))))))))