2-ancestry mixing, positive discriminant

Percentage Accurate: 44.5% → 95.7%

Time: 13.9s

Alternatives: 4

Speedup: 4.2×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 95.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.3%

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

3. Simplified 45.3%

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

5. Taylor expanded in g around inf 72.9%

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

   1. pow1/3 34.5%

      \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]

   2. div-inv 34.5%

      \[\leadsto {\color{blue}{\left(g \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]

   3. unpow-prod-down 25.6%

      \[\leadsto \color{blue}{\left({g}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]

   4. pow1/3 47.2%

      \[\leadsto \left(\color{blue}{\sqrt[3]{g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]

7. Applied egg-rr 47.2%

   \[\leadsto \color{blue}{\left(\sqrt[3]{g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]
8. Step-by-step derivation

   1. unpow1/3 94.7%

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

9. Simplified 94.7%

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

11. Applied egg-rr 95.4%

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

    1. associate-/r/ 95.4%

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

13. Simplified 95.4%

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

    1. associate-*l/ 95.4%

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

15. Applied egg-rr 95.4%

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

    1. mul-1-neg 95.4%

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

17. Simplified 95.4%

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

18. Final simplification 95.4%

    \[\leadsto \frac{\sqrt[3]{g}}{-\sqrt[3]{a}} \]
19. Add Preprocessing

Alternative 2: 74.2% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.3%

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

3. Simplified 45.3%

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

5. Taylor expanded in g around inf 72.9%

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

   1. pow1 72.9%

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

   2. add-cbrt-cube 72.8%

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

   3. pow3 72.8%

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

   4. unpow-prod-down 72.8%

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

   5. pow3 72.9%

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

   6. add-cube-cbrt 72.9%

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

   7. pow3 72.9%

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

   8. cbrt-unprod 73.1%

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

   9. cbrt-unprod 73.4%

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

   10. cbrt-unprod 73.6%

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

   11. rem-3cbrt-lft 73.6%

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

   12. metadata-eval 73.6%

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

7. Applied egg-rr 73.6%

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

   1. unpow1 73.6%

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

   2. *-commutative 73.6%

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

   3. neg-mul-1 73.6%

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

   4. distribute-neg-frac2 73.6%

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

9. Simplified 73.6%

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

10. Final simplification 73.6%

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

Alternative 3: 1.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.3%

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

3. Simplified 45.3%

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

5. Taylor expanded in g around inf 72.9%

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

   1. pow1 72.9%

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

   2. add-cbrt-cube 72.8%

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

   3. pow3 72.8%

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

   4. unpow-prod-down 72.8%

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

   5. pow3 72.9%

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

   6. add-cube-cbrt 72.9%

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

   7. pow3 72.9%

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

   8. cbrt-unprod 73.1%

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

   9. cbrt-unprod 73.4%

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

   10. cbrt-unprod 73.6%

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

   11. rem-3cbrt-lft 73.6%

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

   12. metadata-eval 73.6%

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

7. Applied egg-rr 73.6%

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

   1. unpow1 73.6%

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

   2. *-commutative 73.6%

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

   3. neg-mul-1 73.6%

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

   4. distribute-neg-frac2 73.6%

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

9. Simplified 73.6%

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

11. Applied egg-rr 0.5%

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

    1. times-frac 0.5%

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

    2. rem-square-sqrt 0.9%

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

    3. rem-square-sqrt 1.7%

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

13. Simplified 1.7%

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

Alternative 4: 1.3% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.3%

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

3. Simplified 45.3%

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

5. Taylor expanded in g around inf 72.9%

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

   1. pow1/3 34.5%

      \[\leadsto \color{blue}{{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]

   2. div-inv 34.5%

      \[\leadsto {\color{blue}{\left(g \cdot \frac{1}{a}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]

   3. unpow-prod-down 25.6%

      \[\leadsto \color{blue}{\left({g}^{0.3333333333333333} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]

   4. pow1/3 47.2%

      \[\leadsto \left(\color{blue}{\sqrt[3]{g}} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right) \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]

7. Applied egg-rr 47.2%

   \[\leadsto \color{blue}{\left(\sqrt[3]{g} \cdot {\left(\frac{1}{a}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]
8. Step-by-step derivation

   1. unpow1/3 94.7%

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

9. Simplified 94.7%

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

11. Applied egg-rr 1.3%

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

    1. rem-cube-cbrt 1.3%

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

13. Simplified 1.3%

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

Reproduce

herbie shell --seed 2024153 
(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))))))))