2-ancestry mixing, positive discriminant

Percentage Accurate: 43.9% → 95.8%

Time: 8.2s

Alternatives: 5

Speedup: 3.9×

Specification

Math

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

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: 43.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 95.8% accurate, 2.0× speedup

Math

\[\sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + g} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.9%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.1

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

4. Applied rewrites 95.1%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. pow1/3 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. cbrt-unprod N/A

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

   11. metadata-eval N/A

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

   12. pow1/3 N/A

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

   13. pow-prod-down N/A

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

   14. *-commutative N/A

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

   15. mul-1-neg N/A

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

   16. lift-neg.f64 N/A

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

   17. pow1/3 N/A

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

   18. lift-neg.f64 N/A

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

   19. mul-1-neg N/A

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

   20. *-commutative N/A

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

   21. *-commutative N/A

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

   22. mul-1-neg N/A

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

   23. cbrt-neg-rev N/A

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

   24. lift-cbrt.f64 N/A

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

6. Applied rewrites 73.5%

   \[\leadsto -\sqrt[3]{\frac{g}{a}} \]
7. Step-by-step derivation

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lower-pow.f64 35.1

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

8. Applied rewrites 35.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow1/3 N/A

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

   4. lift-/.f64 N/A

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

   5. cbrt-div N/A

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

   6. *-rgt-identity N/A

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

   7. metadata-eval N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. cbrt-neg-rev N/A

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

   13. cbrt-unprod N/A

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

   14. lift-cbrt.f64 N/A

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

   15. lift-cbrt.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-cbrt.f64 N/A

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

   18. distribute-frac-neg2 N/A

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

   19. frac-2neg N/A

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

   20. lift-*.f64 N/A

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

   21. associate-/l* N/A

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

10. Applied rewrites 95.7%

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

Alternative 2: 95.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.9%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.1

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

4. Applied rewrites 95.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. cbrt-unprod N/A

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

   9. metadata-eval N/A

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

   10. lift-cbrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. cbrt-neg-rev N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lift-cbrt.f64 N/A

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

   17. lower-/.f64 95.7

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

6. Applied rewrites 95.7%

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

Alternative 3: 95.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.9%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.1

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

4. Applied rewrites 95.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. cbrt-unprod N/A

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

   9. metadata-eval N/A

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

   10. lift-cbrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. cbrt-neg-rev N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lift-cbrt.f64 N/A

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

   17. lower-/.f64 95.7

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

6. Applied rewrites 95.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. mult-flip-rev N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 95.8

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

8. Applied rewrites 95.8%

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

Alternative 4: 74.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.9%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.1

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

4. Applied rewrites 95.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. cbrt-unprod N/A

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

   9. metadata-eval N/A

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

   10. lift-cbrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. cbrt-neg-rev N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lift-cbrt.f64 N/A

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

   17. lower-/.f64 95.7

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

6. Applied rewrites 95.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. mul-1-neg N/A

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

   5. lift-cbrt.f64 N/A

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

   6. cbrt-neg-rev N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. cbrt-div N/A

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

   10. div-flip-rev N/A

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

   11. lift-/.f64 N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. cbrt-div N/A

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

   15. metadata-eval N/A

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

   16. cbrt-neg-rev N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. lower-/.f64 N/A

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

   20. lower-cbrt.f64 N/A

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

   21. lift-/.f64 N/A

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

   22. distribute-neg-frac N/A

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

   23. lift-neg.f64 N/A

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

   24. frac-2neg N/A

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

   25. lower-/.f64 74.1

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

8. Applied rewrites 74.1%

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

Alternative 5: 73.5% accurate, 3.9× speedup

Math

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

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 Float64(-cbrt(Float64(g / a)))
end

Wolfram

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

Derivation

1. Initial program 43.9%

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

2. Taylor expanded in g around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-cbrt.f64 95.1

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

4. Applied rewrites 95.1%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. pow1/3 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. cbrt-unprod N/A

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

   11. metadata-eval N/A

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

   12. pow1/3 N/A

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

   13. pow-prod-down N/A

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

   14. *-commutative N/A

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

   15. mul-1-neg N/A

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

   16. lift-neg.f64 N/A

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

   17. pow1/3 N/A

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

   18. lift-neg.f64 N/A

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

   19. mul-1-neg N/A

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

   20. *-commutative N/A

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

   21. *-commutative N/A

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

   22. mul-1-neg N/A

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

   23. cbrt-neg-rev N/A

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

   24. lift-cbrt.f64 N/A

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

6. Applied rewrites 73.5%

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

Reproduce

herbie shell --seed 2025175 
(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))))))))