2-ancestry mixing, positive discriminant

Percentage Accurate: 44.0% → 95.9%

Time: 18.3s

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: 44.0% 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 41.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. neg-sub0 N/A

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   10. cbrt-lowering-cbrt.f64 72.2%

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

5. Simplified 72.2%

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

   1. associate-*l* N/A

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

   2. cbrt-div N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

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

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt[3]{g} \cdot 1\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{g}\right), 1\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), 1\right), \left(\sqrt[3]{a}\right)\right)\right) \]

   10. cbrt-lowering-cbrt.f64 96.4%

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

7. Applied egg-rr 96.4%

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

   1. sub0-neg N/A

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

   2. distribute-neg-frac N/A

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

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

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

   4. neg-sub0 N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{g} \cdot 1\right)\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right) \]

   6. *-rgt-identity N/A

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

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

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

   8. cbrt-lowering-cbrt.f64 96.4%

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

9. Applied egg-rr 96.4%

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

    1. sub0-neg N/A

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

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

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

    3. cbrt-lowering-cbrt.f64 96.4%

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

11. Applied egg-rr 96.4%

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

12. Final simplification 96.4%

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

Alternative 2: 73.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. neg-sub0 N/A

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   10. cbrt-lowering-cbrt.f64 72.2%

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

5. Simplified 72.2%

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

   1. sub0-neg N/A

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

   2. associate-*l* N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

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

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

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

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

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

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

   12. metadata-eval 72.9%

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

7. Applied egg-rr 72.9%

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

   1. rem-exp-log N/A

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

   2. rem-cube-cbrt N/A

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

   3. rem-exp-log N/A

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

   4. cbrt-div N/A

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

   5. *-rgt-identity N/A

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

   6. frac-2neg N/A

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

   7. div-inv N/A

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

   8. unpow-prod-down N/A

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

   9. cube-neg N/A

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

   10. *-rgt-identity N/A

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

   11. rem-cube-cbrt N/A

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

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

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

   13. neg-sub0 N/A

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

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

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

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

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

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, g\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\sqrt[3]{a}\right)\right)\right), 3\right)\right)\right), -1\right) \]

   17. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, g\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(0 - \sqrt[3]{a}\right)\right), 3\right)\right)\right), -1\right) \]

   18. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, g\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\sqrt[3]{a}\right)\right)\right), 3\right)\right)\right), -1\right) \]

   19. cbrt-lowering-cbrt.f64 72.7%

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

9. Applied egg-rr 72.7%

   \[\leadsto \sqrt[3]{\color{blue}{\left(0 - g\right) \cdot {\left(\frac{1}{0 - \sqrt[3]{a}}\right)}^{3}}} \cdot -1 \]
10. Step-by-step derivation

    1. sub0-neg N/A

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

    2. cube-div N/A

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

    3. metadata-eval N/A

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

    4. div-inv N/A

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

    5. sub0-neg N/A

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

    6. cube-neg N/A

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

    7. rem-cube-cbrt N/A

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

    8. frac-2neg N/A

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

    9. remove-double-div N/A

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

    10. inv-pow N/A

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

    11. pow-flip N/A

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

    12. metadata-eval N/A

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

    13. metadata-eval N/A

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

    14. pow-prod-up N/A

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

    15. pow2 N/A

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

    16. clear-num N/A

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

    17. div-inv N/A

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

    18. inv-pow N/A

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

    19. associate-/r/ N/A

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

    20. associate-*l* N/A

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

    21. div-inv N/A

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

    22. associate-/l/ N/A

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

    23. associate-*l/ N/A

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

    24. div-inv N/A

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

11. Applied egg-rr 73.0%

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

12. Final simplification 73.0%

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

Alternative 3: 73.6% 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 41.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. neg-sub0 N/A

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   10. cbrt-lowering-cbrt.f64 72.2%

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

5. Simplified 72.2%

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

   1. associate-*l* N/A

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

   2. cbrt-div N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

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

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt[3]{g} \cdot 1\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{g}\right), 1\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), 1\right), \left(\sqrt[3]{a}\right)\right)\right) \]

   10. cbrt-lowering-cbrt.f64 96.4%

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

7. Applied egg-rr 96.4%

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

   1. sub0-neg N/A

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

   2. *-rgt-identity N/A

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

   3. cbrt-div N/A

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

   4. *-rgt-identity N/A

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

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

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

   6. *-rgt-identity N/A

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

   7. rem-exp-log N/A

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

   8. unpow1/3 N/A

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

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

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

   10. rem-exp-log N/A

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

   11. /-lowering-/.f64 36.0%

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

9. Applied egg-rr 36.0%

   \[\leadsto \color{blue}{-{\left(\frac{g}{a}\right)}^{0.3333333333333333}} \]
10. Step-by-step derivation

    1. pow1/3 N/A

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

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

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

    3. /-lowering-/.f64 72.9%

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

11. Applied egg-rr 72.9%

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

12. Final simplification 72.9%

    \[\leadsto 0 - \sqrt[3]{\frac{g}{a}} \]
13. 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 41.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
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. neg-sub0 N/A

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   10. cbrt-lowering-cbrt.f64 72.2%

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

5. Simplified 72.2%

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

   1. associate-*l* N/A

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

   2. cbrt-div N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

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

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt[3]{g} \cdot 1\right), \color{blue}{\left(\sqrt[3]{a}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{g}\right), 1\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(g\right), 1\right), \left(\sqrt[3]{a}\right)\right)\right) \]

   10. cbrt-lowering-cbrt.f64 96.4%

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

7. Applied egg-rr 96.4%

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

   1. sub0-neg N/A

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

   2. distribute-neg-frac N/A

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

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

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

   4. neg-sub0 N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{g} \cdot 1\right)\right), \left(\sqrt[3]{\color{blue}{a}}\right)\right) \]

   6. *-rgt-identity N/A

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

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

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

   8. cbrt-lowering-cbrt.f64 96.4%

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

9. Applied egg-rr 96.4%

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

    1. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(0 - \sqrt[3]{g}\right)\right)}{\color{blue}{\mathsf{neg}\left(\sqrt[3]{a}\right)}} \]

    2. sub0-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(0 - \sqrt[3]{g}\right)\right)}{0 - \color{blue}{\sqrt[3]{a}}} \]

    3. sub0-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt[3]{g}\right)\right)\right)}{0 - \sqrt[3]{a}} \]

    4. remove-double-neg N/A

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

    5. rem-cbrt-cube N/A

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

    6. cbrt-div N/A

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

    7. sub0-neg N/A

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

    8. cube-neg N/A

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

    9. rem-cube-cbrt N/A

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

    10. distribute-neg-frac2 N/A

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

    11. distribute-neg-frac2 N/A

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

    12. rem-cube-cbrt N/A

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

    13. cube-neg N/A

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

    14. sub0-neg N/A

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

    15. frac-2neg N/A

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

    16. cbrt-div N/A

        \[\leadsto \frac{\sqrt[3]{\mathsf{neg}\left(g\right)}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left({\left(0 - \sqrt[3]{a}\right)}^{3}\right)}}} \]

    17. cube-neg N/A

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

    18. sub0-neg N/A

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

    19. remove-double-neg N/A

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

11. Applied egg-rr 1.4%

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

Reproduce

herbie shell --seed 2024138 
(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))))))))