2-ancestry mixing, positive discriminant

Percentage Accurate: 44.8% → 95.9%

Time: 16.5s

Alternatives: 4

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: 44.8% 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: 95.9% 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 48.2%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   7. cbrt-lowering-cbrt.f64 75.8%

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

5. Simplified 75.8%

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

   1. associate-*r* N/A

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

   2. cbrt-div N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

   10. cbrt-lowering-cbrt.f64 95.8%

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

7. Applied egg-rr 95.8%

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

9. Applied egg-rr 95.8%

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

10. Final simplification 95.8%

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

Alternative 2: 74.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.2%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   7. cbrt-lowering-cbrt.f64 75.8%

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

5. Simplified 75.8%

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

   1. associate-*r* N/A

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

   2. cbrt-div N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

   10. cbrt-lowering-cbrt.f64 95.8%

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

7. Applied egg-rr 95.8%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. cbrt-unprod N/A

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

   4. cbrt-undiv N/A

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

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

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

   6. *-commutative N/A

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

   7. neg-mul-1 N/A

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

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

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

   9. neg-sub0 N/A

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

   10. --lowering--.f64 77.8%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(0, g\right)\right)\right)\right) \]

9. Applied egg-rr 77.8%

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

Alternative 3: 74.0% 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 48.2%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   7. cbrt-lowering-cbrt.f64 75.8%

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

5. Simplified 75.8%

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

   1. associate-*r* N/A

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

   2. cbrt-div N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

   10. cbrt-lowering-cbrt.f64 95.8%

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

7. Applied egg-rr 95.8%

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

9. Applied egg-rr 76.5%

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

10. Final simplification 76.5%

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

Alternative 4: 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 48.2%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   7. cbrt-lowering-cbrt.f64 75.8%

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

5. Simplified 75.8%

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

   1. associate-*r* N/A

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

   2. cbrt-div N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

   10. cbrt-lowering-cbrt.f64 95.8%

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

7. Applied egg-rr 95.8%

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

   1. cbrt-unprod N/A

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

   2. cbrt-undiv N/A

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

   3. *-commutative N/A

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

   4. neg-mul-1 N/A

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

   5. distribute-neg-frac N/A

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

   6. sub0-neg N/A

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

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

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

   8. rem-cube-cbrt N/A

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

   9. cube-mult N/A

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

   10. pow1/3 N/A

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

   11. cbrt-unprod N/A

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

   12. pow1/3 N/A

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

   13. pow-prod-down N/A

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

9. Applied egg-rr 1.5%

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

Reproduce

herbie shell --seed 2024163 
(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))))))))