2-ancestry mixing, positive discriminant

Percentage Accurate: 43.4% → 95.7%

Time: 16.9s

Alternatives: 7

Speedup: 4.1×

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: 43.4% 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: 95.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.9%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{2}\right), \color{blue}{\left(\sqrt[3]{\frac{-1}{2}}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}}\right), \left(\sqrt[3]{2}\right)\right), \left(\sqrt[3]{\color{blue}{\frac{-1}{2}}}\right)\right) \]

   5. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right), \left(\sqrt[3]{2}\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \left(\sqrt[3]{2}\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right) \]

   7. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(2\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right) \]

   8. cbrt-lowering-cbrt.f64 71.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(2\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

5. Simplified 71.3%

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

   1. cbrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a} \cdot 2}\right), \mathsf{cbrt.f64}\left(\color{blue}{\frac{-1}{2}}\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\left(g \cdot \frac{1}{a}\right) \cdot 2}\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{g \cdot \left(\frac{1}{a} \cdot 2\right)}\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   4. cbrt-prod N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a} \cdot 2}\right), \mathsf{cbrt.f64}\left(\color{blue}{\frac{-1}{2}}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{g}\right), \left(\sqrt[3]{\frac{1}{a} \cdot 2}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{\frac{-1}{2}}\right)\right) \]

   6. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \left(\sqrt[3]{\frac{1}{a} \cdot 2}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   7. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\left(\frac{1}{a} \cdot 2\right)\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{a}\right), 2\right)\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   9. /-lowering-/.f64 96.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, a\right), 2\right)\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

7. Applied egg-rr 96.0%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{a} \cdot 2} \cdot \sqrt[3]{g}\right), \mathsf{cbrt.f64}\left(\color{blue}{\frac{-1}{2}}\right)\right) \]

   2. pow1/3 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{a} \cdot 2} \cdot {g}^{\frac{1}{3}}\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   3. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{a} \cdot 2} \cdot \left({g}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {g}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   4. pow-prod-down N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{a} \cdot 2} \cdot {\left(g \cdot g\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   5. sqr-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{a} \cdot 2} \cdot {\left(\left(\mathsf{neg}\left(g\right)\right) \cdot \left(\mathsf{neg}\left(g\right)\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   6. sub0-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{a} \cdot 2} \cdot {\left(\left(0 - g\right) \cdot \left(\mathsf{neg}\left(g\right)\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   7. sub0-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{a} \cdot 2} \cdot {\left(\left(0 - g\right) \cdot \left(0 - g\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   8. pow-prod-down N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{a} \cdot 2} \cdot \left({\left(0 - g\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {\left(0 - g\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   9. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{a} \cdot 2} \cdot {\left(0 - g\right)}^{\frac{1}{3}}\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{a} \cdot 2}\right), \left({\left(0 - g\right)}^{\frac{1}{3}}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{\frac{-1}{2}}\right)\right) \]

   11. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{a} \cdot 2\right)\right), \left({\left(0 - g\right)}^{\frac{1}{3}}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   12. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1 \cdot 2}{a}\right)\right), \left({\left(0 - g\right)}^{\frac{1}{3}}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{2}{a}\right)\right), \left({\left(0 - g\right)}^{\frac{1}{3}}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \left({\left(0 - g\right)}^{\frac{1}{3}}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   15. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \left({\left(0 - g\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {\left(0 - g\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   16. pow-prod-down N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \left({\left(\left(0 - g\right) \cdot \left(0 - g\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   17. sub0-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \left({\left(\left(\mathsf{neg}\left(g\right)\right) \cdot \left(0 - g\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   18. sub0-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \left({\left(\left(\mathsf{neg}\left(g\right)\right) \cdot \left(\mathsf{neg}\left(g\right)\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   19. sqr-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \left({\left(g \cdot g\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   20. pow-prod-down N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \left({g}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {g}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   21. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \left({g}^{\frac{1}{3}}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   22. pow1/3 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \left(\sqrt[3]{g}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   23. cbrt-lowering-cbrt.f64 96.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \mathsf{cbrt.f64}\left(g\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

9. Applied egg-rr 96.0%

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

Alternative 2: 95.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{2}{a}} \cdot \sqrt[3]{g \cdot -0.5} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.9%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{2}\right), \color{blue}{\left(\sqrt[3]{\frac{-1}{2}}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}}\right), \left(\sqrt[3]{2}\right)\right), \left(\sqrt[3]{\color{blue}{\frac{-1}{2}}}\right)\right) \]

   5. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right), \left(\sqrt[3]{2}\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \left(\sqrt[3]{2}\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right) \]

   7. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(2\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right) \]

   8. cbrt-lowering-cbrt.f64 71.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(2\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

5. Simplified 71.3%

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

   1. cbrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a} \cdot 2}\right), \mathsf{cbrt.f64}\left(\color{blue}{\frac{-1}{2}}\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\left(g \cdot \frac{1}{a}\right) \cdot 2}\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{g \cdot \left(\frac{1}{a} \cdot 2\right)}\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   4. cbrt-prod N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a} \cdot 2}\right), \mathsf{cbrt.f64}\left(\color{blue}{\frac{-1}{2}}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{g}\right), \left(\sqrt[3]{\frac{1}{a} \cdot 2}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{\frac{-1}{2}}\right)\right) \]

   6. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \left(\sqrt[3]{\frac{1}{a} \cdot 2}\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   7. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\left(\frac{1}{a} \cdot 2\right)\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{a}\right), 2\right)\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

   9. /-lowering-/.f64 96.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, a\right), 2\right)\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

7. Applied egg-rr 96.0%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{a} \cdot 2}\right), \color{blue}{\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{-1}{2}}\right)}\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{a} \cdot 2\right)\right), \left(\color{blue}{\sqrt[3]{g}} \cdot \sqrt[3]{\frac{-1}{2}}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1 \cdot 2}{a}\right)\right), \left(\sqrt[3]{\color{blue}{g}} \cdot \sqrt[3]{\frac{-1}{2}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{2}{a}\right)\right), \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{-1}{2}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \left(\sqrt[3]{\color{blue}{g}} \cdot \sqrt[3]{\frac{-1}{2}}\right)\right) \]

   8. cbrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \left(\sqrt[3]{g \cdot \frac{-1}{2}}\right)\right) \]

   9. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \mathsf{cbrt.f64}\left(\left(g \cdot \frac{-1}{2}\right)\right)\right) \]

   10. *-lowering-*.f64 96.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(2, a\right)\right), \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(g, \frac{-1}{2}\right)\right)\right) \]

9. Applied egg-rr 96.0%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{2}{a}} \cdot \sqrt[3]{g \cdot -0.5}} \]
10. Add Preprocessing

Alternative 3: 95.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.9%

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

3. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}}\right), \color{blue}{\left(\mathsf{neg}\left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)\right)}\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \left(\mathsf{neg}\left(\color{blue}{\sqrt[3]{\frac{1}{2}}} \cdot \sqrt[3]{2}\right)\right)\right) \]

   6. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{neg.f64}\left(\left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{2}}\right), \left(\sqrt[3]{2}\right)\right)\right)\right) \]

   8. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\frac{1}{2}\right), \left(\sqrt[3]{2}\right)\right)\right)\right) \]

   9. cbrt-lowering-cbrt.f64 71.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\frac{1}{2}\right), \mathsf{cbrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 71.5%

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

   1. cbrt-div N/A

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

   2. cbrt-unprod N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{g} \cdot -1\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{g}\right), -1\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right) \]

   9. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), -1\right), \left(\sqrt[3]{a}\right)\right) \]

   10. cbrt-lowering-cbrt.f64 95.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), -1\right), \mathsf{cbrt.f64}\left(a\right)\right) \]

7. Applied egg-rr 95.9%

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

8. Final simplification 95.9%

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

Alternative 4: 74.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt[3]{\frac{0 - a}{g}}} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (/ 1.0 (cbrt (/ (- 0.0 a) g))))

C

double code(double g, double h, double a) {
	return 1.0 / cbrt(((0.0 - a) / g));
}

Java

public static double code(double g, double h, double a) {
	return 1.0 / Math.cbrt(((0.0 - a) / g));
}

Julia

function code(g, h, a)
	return Float64(1.0 / cbrt(Float64(Float64(0.0 - a) / g)))
end

Wolfram

code[g_, h_, a_] := N[(1.0 / N[Power[N[(N[(0.0 - a), $MachinePrecision] / g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.9%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{2}\right), \color{blue}{\left(\sqrt[3]{\frac{-1}{2}}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}}\right), \left(\sqrt[3]{2}\right)\right), \left(\sqrt[3]{\color{blue}{\frac{-1}{2}}}\right)\right) \]

   5. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right), \left(\sqrt[3]{2}\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \left(\sqrt[3]{2}\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right) \]

   7. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(2\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right) \]

   8. cbrt-lowering-cbrt.f64 71.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(2\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

5. Simplified 71.3%

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

   1. associate-*l* N/A

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

   2. pow1/3 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{2} \cdot {\frac{1}{4}}^{\left(\frac{\color{blue}{\frac{1}{3}}}{2}\right)}\right) \]

   6. metadata-eval N/A

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

   7. pow-prod-down N/A

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

   8. sqr-pow N/A

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

   9. pow1/3 N/A

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

   10. *-commutative N/A

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

   11. cbrt-unprod N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. clear-num N/A

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

   16. frac-2neg N/A

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

   17. metadata-eval N/A

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

   18. cbrt-div N/A

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

7. Applied egg-rr 72.8%

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

8. Final simplification 72.8%

   \[\leadsto \frac{1}{\sqrt[3]{\frac{0 - a}{g}}} \]
9. Add Preprocessing

Alternative 5: 34.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ 0 - {\left(\frac{g}{a}\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (- 0.0 (pow (/ g a) 0.3333333333333333)))

C

double code(double g, double h, double a) {
	return 0.0 - pow((g / a), 0.3333333333333333);
}

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = 0.0d0 - ((g / a) ** 0.3333333333333333d0)
end function

Java

public static double code(double g, double h, double a) {
	return 0.0 - Math.pow((g / a), 0.3333333333333333);
}

Python

def code(g, h, a):
	return 0.0 - math.pow((g / a), 0.3333333333333333)

Julia

function code(g, h, a)
	return Float64(0.0 - (Float64(g / a) ^ 0.3333333333333333))
end

MATLAB

function tmp = code(g, h, a)
	tmp = 0.0 - ((g / a) ^ 0.3333333333333333);
end

Wolfram

code[g_, h_, a_] := N[(0.0 - N[Power[N[(g / a), $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.9%

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

3. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}}\right), \color{blue}{\left(\mathsf{neg}\left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)\right)}\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \left(\mathsf{neg}\left(\color{blue}{\sqrt[3]{\frac{1}{2}}} \cdot \sqrt[3]{2}\right)\right)\right) \]

   6. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{neg.f64}\left(\left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{2}}\right), \left(\sqrt[3]{2}\right)\right)\right)\right) \]

   8. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\frac{1}{2}\right), \left(\sqrt[3]{2}\right)\right)\right)\right) \]

   9. cbrt-lowering-cbrt.f64 71.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\frac{1}{2}\right), \mathsf{cbrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 71.5%

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

   1. cbrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. *-rgt-identity N/A

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

   6. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt[3]{\frac{g}{a}}\right)\right) \]

   7. pow1/3 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{g}{a}\right)}^{\frac{1}{3}}\right)\right) \]

   8. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\left(\frac{g}{a}\right), \frac{1}{3}\right)\right) \]

   9. /-lowering-/.f64 33.2%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(g, a\right), \frac{1}{3}\right)\right) \]

7. Applied egg-rr 33.2%

   \[\leadsto \color{blue}{-{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \]

8. Final simplification 33.2%

   \[\leadsto 0 - {\left(\frac{g}{a}\right)}^{0.3333333333333333} \]
9. Add Preprocessing

Alternative 6: 73.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.9%

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) + \sqrt{g \cdot g - h \cdot h}\right)}\right), \color{blue}{\left(\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(\mathsf{neg}\left(g\right)\right) - \sqrt{g \cdot g - h \cdot h}\right)}\right)}\right) \]

3. Simplified 39.9%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a \cdot -2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. cbrt-div N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}}\right), \mathsf{cbrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{\frac{\sqrt[3]{2 \cdot a}}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}}\right), \mathsf{cbrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)}\right)\right) \]

   3. cbrt-div N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{\sqrt[3]{\frac{2 \cdot a}{\sqrt{g \cdot g - h \cdot h} - g}}}\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \color{blue}{\mathsf{*.f64}\left(a, -2\right)}\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\sqrt[3]{\frac{2 \cdot a}{\sqrt{g \cdot g - h \cdot h} - g}}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)}\right)\right) \]

   5. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{2 \cdot a}{\sqrt{g \cdot g - h \cdot h} - g}\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \color{blue}{\mathsf{*.f64}\left(a, -2\right)}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot a\right), \left(\sqrt{g \cdot g - h \cdot h} - g\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{a}, -2\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(a \cdot 2\right), \left(\sqrt{g \cdot g - h \cdot h} - g\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{1}{\frac{1}{2}}\right), \left(\sqrt{g \cdot g - h \cdot h} - g\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)\right)\right) \]

   9. div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{a}{\frac{1}{2}}\right), \left(\sqrt{g \cdot g - h \cdot h} - g\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(a, \frac{1}{2}\right), \left(\sqrt{g \cdot g - h \cdot h} - g\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(a, \frac{1}{2}\right), \mathsf{\_.f64}\left(\left(\sqrt{g \cdot g - h \cdot h}\right), g\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)\right)\right) \]

   12. rem-square-sqrt N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(a, \frac{1}{2}\right), \mathsf{\_.f64}\left(\left(\sqrt{\sqrt{g \cdot g - h \cdot h} \cdot \sqrt{g \cdot g - h \cdot h}}\right), g\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)\right)\right) \]

   13. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(a, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\sqrt{g \cdot g - h \cdot h} \cdot \sqrt{g \cdot g - h \cdot h}\right)\right), g\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)\right)\right) \]

   14. rem-square-sqrt N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(a, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(g \cdot g - h \cdot h\right)\right), g\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)\right)\right) \]

   15. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(a, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(g \cdot g\right), \left(h \cdot h\right)\right)\right), g\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(a, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \left(h \cdot h\right)\right)\right), g\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)\right)\right) \]

   17. *-lowering-*.f64 40.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(a, \frac{1}{2}\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right), g\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(g, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(g, g\right), \mathsf{*.f64}\left(h, h\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right)\right)\right) \]

6. Applied egg-rr 40.8%

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

7. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\sqrt[3]{\frac{g}{a}}\right)}\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right)\right) \]

   5. /-lowering-/.f64 72.1%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right)\right) \]

9. Simplified 72.1%

   \[\leadsto \color{blue}{0 - \sqrt[3]{\frac{g}{a}}} \]
10. Add Preprocessing

Alternative 7: 1.4% 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(g / a))
end

Wolfram

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

Derivation

1. Initial program 39.9%

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

3. Taylor expanded in g around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{2}\right), \color{blue}{\left(\sqrt[3]{\frac{-1}{2}}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{\frac{g}{a}}\right), \left(\sqrt[3]{2}\right)\right), \left(\sqrt[3]{\color{blue}{\frac{-1}{2}}}\right)\right) \]

   5. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right), \left(\sqrt[3]{2}\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \left(\sqrt[3]{2}\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right) \]

   7. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(2\right)\right), \left(\sqrt[3]{\frac{-1}{2}}\right)\right) \]

   8. cbrt-lowering-cbrt.f64 71.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right), \mathsf{cbrt.f64}\left(2\right)\right), \mathsf{cbrt.f64}\left(\frac{-1}{2}\right)\right) \]

5. Simplified 71.3%

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

   1. associate-*l* N/A

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

   2. pow1/3 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{2} \cdot {\frac{1}{4}}^{\left(\frac{\color{blue}{\frac{1}{3}}}{2}\right)}\right) \]

   6. metadata-eval N/A

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

   7. pow-prod-down N/A

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

   8. sqr-pow N/A

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

   9. pow1/3 N/A

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

   10. *-commutative N/A

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

   11. cbrt-unprod N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{cbrt.f64}\left(\left(\frac{g}{a}\right)\right) \]

   16. /-lowering-/.f64 1.4%

       \[\leadsto \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(g, a\right)\right) \]

7. Applied egg-rr 1.4%

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

Reproduce

herbie shell --seed 2024191 
(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))))))))